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unique perfect phylogeny characterizations via uniquely representable chordal graphs may rob gysel department computer science university california davis shields avenue davis usa rsgysel abstract perfect phylogeny problem classic problem computational biology seek unrooted phylogeny compatible set qualitative characters tree exists precisely intersection graph associated character set called partition intersection graph triangulated using restricted set fill edges semple steel used partition intersection graph characterize character set unique perfect phylogeny bordewich huber semple showed use partition intersection graph find maximum compatible set characters paper build results characterizing unique perfect phylogeny exists subset partial characters characterization stated terms minimal triangulations partition intersection graph uniquely representable also known graphs characterization motivated structure graphs fact block structure minimal triangulations mirrored graph triangulated introduction pair tree map nodes every node degree two one mapped call range labeled nodes nodes labeled underlying tree free bijection leaves ternary every internal node degree three given use denote minimal subtree containing nodes two subtrees intersect one nodes common common node say intersect partial character partition subset called cell intersect every pair distinct cells displays perfect phylogeny set partial characters displays character perfect phylogeny also say compatible perfect phylogeny problem also called character compatibility problem determine set partial characters compatible bde int fig partition intersection graph int characters edges int given solid edges dashed edge represents fill edge required obtain triangulation int int displayed bde intersect intersection induces dashed fill edge int breaks edge distinguished perfect phylogeny problem reduces graph theoretical problem detail given set characters one construct partition intersection graph int follows vertex set int cell edge two vertices intersection vertex int cell character observe partial character every pair distinct vertices int graph chordal every cycle length four chord edge vertices cycle appear consecutively cycle general int chordal graph interested adding edges int obtain chordal supergraph int called triangulation int added edges called fill edges subset fill edges yields triangulation int minimal triangulation int fill edge form resulting triangulation proper triangulation int following classic result reduces question determining compatibility finding proper triangulations partition intersection graph originally phrased terms proper triangulations definitions follows int proper triangulation proper minimal triangulation theorem let set qualitative characters compatible int proper minimal triangulation bijective two isomorphic writing map following properties preserves labels meaning graph isomorphism set characters defines perfect phylogeny unique perfect phylogeny isomorphism unique perfect phylogeny problem determine set partial characters defines perfect phylogeny displays edge node node distinct cells distinguished every edge distinguished least one character distinguished following characterization due semple steel theorem let set partial characters defines perfect phylogeny following conditions satisfied int unique proper minimal triangulation free ternary perfect phylogeny distinguished unique perfect phylogeny free ternary distinguished int result impetus current work one main interests condition terms combinatorial structures play significant role study chordal graphs minimal triangulations chordal graphs characterized existence trees represent adjacency structure graph suppose graph vertices tree representation consists tree subtrees two trees intersect adjacent subtrees correspondence vertex set correspondence made explicit mapping subtree vertex observe node defines clique node notationally write tree representation ordered pair maps nodes cliques satisfying following properties edge coverage pair vertices adjacent node convexity vertex set nodes induces subtree connected subgraph frequently refer convexity property throughout paper call underlying tree often define tree representation specifying underlying tree collection subtrees together implicitly define let collection subtrees pair subtrees intersect node vij helly property subtrees tree intersect common node property manifests statement cliques nodes following way clique least one node particular true maximal clique proper superset also clique therefore contains set maximal cliques maximal cliques correspondence nodes via clique tree see figure example theorem following statements equivalent chordal graph tree representation clique tree observe clique tree edge maximal cliques vertex general chordal graph exponential number clique trees algorithm enumerate clique trees along formula count appears often analyzing tree representation triangulation int given vertex int denote subtree corresponds observe node given set characters chordal graph int given adding edge two vertices intersect construction along fact int triangulation int phylogenetics literature discussed detail section chordal graph uniquely representable single clique tree short graph ternary internal node clique tree degree three leafage number leaves clique tree let proper triangulation int clique tree edge incontractable respect distinct cells say incontractable respect edge incontractable respect least one present first main result theorem suppose set partial characters defines perfect phylogeny following conditions hold int unique proper minimal triangulation ternary graph leafage edge unique clique tree incontractable respect perfect phylogeny defined free ternary distinguished int let set partial characters suppose triangulation int fill edge distinct cells general leafage chordal graph minimum number leaves clique tree graph say breaks broken character triangulation int displayed characters characters broken characters bordewich huber semple proved possible find compatible subset using partition intersection graph theorem let set partial characters maximumsized compatible subset triangulation int displayed characters triangulation int displayed characters subset partial characters maximal defining subset defines perfect phylogeny compatible set second main result following theorem suppose set partial characters maximal defining subset following conditions hold int unique minimal triangulation displayed characters minimal triangulation int least displayed character set ternary graph leafage edge unique clique tree incontractable respect perfect phylogeny defined free ternary distinguished int chordal graph preliminaries section detail known results chordal graphs necessary remainder paper suppose graph let denote graph obtained removing edges incident least one vertex vertices connected proper subset property minimal minimal least one pair vertices minimal separator minimality definition relative possible containment relationships two minimal maximal connected subsets connected components let connected component neighborhoood denoted set vertices adjacent least one vertex full component following useful characterization minimal separators left exercise dirac called sets relatively minimal perhaps descriptive term stuck modern literature chordal graphs minimal triangulations fig chordal graph clique tree three clique trees one obtained removing node incident edge attaching maximal clique map defined triangle drawn inside node arrows indicate vertices consist intersection neighboring node maximal clique intersection minimal separator theorem lemma let graph minimal separator two full components minimal separator multiplicity full components interestingly clique trees contain detailed information minimal separators graph represents useful proofs later sections theorem suppose chordal graph clique tree minimal separator edge multiplicity number edges property need following characterizations graphs list completeness theorem let chordal graph following statements equivalent uniquely representable minimal separator exactly two maximal cliques minimal separator multiplicity one theorem also implies minimal separator chordal graph clique fact chordal graphs characterized clique minimal separators one earliest results chordal graphs minimal separator properly contains another minimal separator number minimal separators number maximal cliques minus one chordal graphs recognized linear time maximal cliques minimal separators chordal graph may computed linear time using property theorem allows recognition graphs linear time clique trees tree representations section define two operations facilitate discussion tree representations prove state results useful later sections operations commonly used literature see seem named results section useful proving characterization maximal defining subsets given set characters construct chordal graph int vertex set identical int edge intersect graph tree representation underlying tree subtrees obtained defining vertex int therefore int chordal theorem edge int cells share least one member say intersect therefore intersect well edge int subtree intersection hence edge int also edge int int triangulation int say derives tree representation int derived general clique tree observation let set partial characters tree representation induced underlying tree underlying tree cells int triangulation int lemma let set partial characters displays suppose edge distinguished tree representation int derived clique tree int proof let tree representation int derived sake contradiction assume clique tree map nodes maximal cliques int must nodes let bde fig tree representation int derived figure triangulation int int depicted figure note two subtrees intersect precisely corresponding vertices int adjacent observe clique tree example two nodes map nonmaximal cliques additionally two nodes map maximal clique bde obtaining clique tree tree representation described closest node allowing possibility note vertex also vertex convexity distinguished distinct cells node node observation node containment fill edge int breaks contradicts assumption displays must clique tree int lemma let set partial characters displays suppose free ternary edge distinguished int uniquely representable proof prove int uniquely representable use theorem show containment relationships minimal separators int working towards contradiction assume minimal separators int let tree representation derived clique tree lemma edges theorem without loss generality assume path contain either perhaps let node path adjacent otherwise let either case contains otherwise element convexity complete proof obtain contradiction showing vertex character distinguishes distinct cells node node least two neighbors ternary must degree three also mapped free order node must least two nodes path two nodes must contain path must also contain two neighbors neither vertices node thus must path node contain observation must vertex node impossible shown thus containment relationships minimal separators int int uniquely representable theorem lemma let set partial characters subset displayed int triangulation int displayed characters previous three lemmas summarized follows theorem let set partial characters displays suppose free ternary edge distinguished int uniquely representable chordal graph tree representation derived unique clique tree displayed characters int exactly suppose clique tree triangulation int goal defining discussion provide standard see construct map defining contains every vertex int whose cell contains vertices form clique contained cells added fill edges obtain clique subset maximal clique hence exists may one choice call candidate node let leaf neighbor maximal clique contains vertex int found convexity node whose corresponding maximal clique contains unique candidate node hence every leaf unique candidate node least one element thus leaf must labeled finish constructing obtain suppressing unlabeled nodes degree two result say induces induced emphasize underlying tree need underlying tree note element may multiple candidate nodes may induce multiple next show minimal triangulation much structure described following lemma useful lemma let graph minimal triangulation fill edge minimal separator contains fig clique tree int figure induced note int int fig though stated form following lemma follows proof lemma statement corollary lemma let minimal triangulation int suppose induced clique tree int lemma let minimal triangulation int clique tree suppose induces underlying tree proof already seen every leaf unique candidate node element addition also shown node degree two unique candidate node element done showing contains either vertex int contained node edge int whose incident vertices cells intersection contained node completeness outline proof using convexity fact degree two follows either unique vertex unique pair vertices contained remains show actually edge int lemma minimal separator containing theorem edge contradicts case must edge int cases unique candidate node element element either every degree two node labeled nodes need suppressed hence underlying tree lemma let minimal triangulation int clique tree suppose induces vertex int proof let vertex int consider node either lies first case candidate node similarly second case element therefore convexity cases node finish proving equality suppose define tree representation graph follows set define subtrees vertex int follows already seen every vertex int edge set subset edge set edge int least one element common thus intersect vertex int also intersect therefore edge chordal theorem triangulation int complete proof show must edge exist node adjacent node maximality vertex node therefore edge situation different edge intersect node node path convexity would imply hence edge edge set proper subset edge set impossible minimal triangulation int must vertex int lemmas summarized theorem let minimal triangulation int clique tree suppose induces underlying tree underlying tree vertex int int maximal defining subsets characters section devoted proof theorem proof follow mainly propositions recall graph subset vertices graph obtained removing vertices edges incident one vertices lemma let set partial characters suppose minimal triangulation int let vertices int vertices int minimal triangulation int proof let minimal triangulation int graph obtained adding following fill edges int fill edges fill edges form vertex vertex int first prove chordal use construct minimal triangulation int let cycle cycle also cycle therefore chord edge edge also edge cycle chord otherwise without loss generality either edge int fill edge type either case cycle chord chordal let minimal triangulation int edge set subset edge set every edge either edge int edge construction therefore edge set subset edge set chordal cycle cycle chordality inherited triangulation int minimality must edge set equal edge set lemma see also let set partial characters suppose compatible subset minimal triangulation int whose displayed characters least though stated form following lemma direct result lemma proof lemma let set partial characters suppose triangulation int displayed characters induced clique tree perfect phylogeny proposition let set partial characters suppose following conditions hold int unique minimal triangulation displayed characters minimal triangulation int least displayed character set ternary graph leafage iii edge unique clique tree incontractable respect maximal defining subset proof begin showing unique perfect phylogeny using theorem finish proof showing superset unique perfect phylogeny throughout proof denote unique minimal triangulation int given whose displayed character set lemma perfect phylogeny compatible proper triangulation int theorem int proper minimal triangulation well see condition theorem holds suppose proper minimal triangulations int given theorem let set vertices int int lemma minimal triangulations int satisfy displayed characters must least fill edge breaks character would also appear proper triangulations int therefore condition theorem satisfied respect show condition theorem holds let unique clique tree given suppose induced displays lemma theorem underlying tree also underlying tree ternary leaves must free ternary see distinguished consider edge iii incontractible respect character distinct cells node node theorem distinguishes hence condition theorem also holds respect defines last show proper superset also defines superset compatible lemma minimal triangulation int least displayed character set would contradict superset exist completes proof lemma suppose maximal defining subset minimal triangulations int displayed character set proof let induced clique tree induced clique tree lemma perfect phylogenies defines must via isomorphism additionally vertex int theorem prove suffices show fill edge sets suppose fill edge edge coverage property clique trees intersect node show intersect node otherwise internal node path graph isomorphism internal node path nodes node well cases node similar argument shows node therefore intersect fill edge fill edge set subset fill edge set symmetric argument shows fill edge set subset fill edge set fill edge sets must equal completing proof proposition let set partial characters maximal defining subset following conditions hold int unique minimal triangulation displayed characters minimal triangulation int least displayed character set ternary graph leafage iii edge unique clique tree incontractable respect defines int proof see holds first observe compatible definition minimal triangulation int least displayed character set lemma maximal defining subset compatible lemma displayed characters compatible set must exactly true minimal triangulation int least displayed characters minimal triangulation lemma unique minimal triangulation int least displayed characters refer unique minimal triangulation remainder proof show holds let induced clique tree lemma perfect phylogeny since maximal defining subset must defines recall free ternary distinguished according theorem theorem int urchordal hand int theorem well theorem unique clique tree underlying tree since ternary clique tree must also ternary ternary graph proves statement consider condition iii let edge theorem edge distinguished character cells node node theorem see incontractable respect hence incontractable respect remainder theorem shown proving holds proof theorem propositions show maximal defining subset conditions hold fact free ternary distinguished follows theorem finally int due proposition proof theorem use theorem taking discussion conclude brief discussion role minimal separators play minimal triangulation theory characterization may contribute towards constructing algorithm sometimes finds maximal defining subset characters one exists minimal triangulations characterized minimal separators happen minimal separators triangulated graph well minimal separator minimal triangulation connected components full components identical graph triangulated todinca used minimal separators potential maximal cliques maximal cliques minimal triangulations create dynamic programming algorithm solve treewidth problems time polynomial number minimal separators graph approach extended create dynamic programming algorithm solves variety perfect phylogeny problems including unique perfect phylogeny problem results elucidate structure minimal separators triangulations associated maximal defining subsets characters structure retained partition intersection graph closely related structure potential maximal cliques connected components obtained removing vertices potential maximal clique neighborhoods minimal separators may allow computation ternary minimal triangulation time polynomial number minimal separators int asserting ternary minimal triangulations exist yielding candidate subset may maximal subset characters number minimal separators int bounded number minimal separators int specific example general fact see corollary therefore computationally feasible find due int small number minimal separators checking defines perfect phylogeny using method may also feasible acknowledgements research partially supported nsf grants references anne berry romain pogorelcnik simple algorithm generate minimal separators maximal cliques chordal graph information processing letters blair peyton introduction chordal graphs clique trees george gilbert liu editors graph theory sparse matrix computations volume ima volumes mathematics applications pages bordewich huber semple identifying phylogenetic trees discrete mathematics todinca treewidth minimum grouping minimal separators siam journal computing todinca listing potential maximal cliques graph theoretical computer science buneman characterisation rigid circuit graphs discrete mathematics dirac rigid circuit graphs abhandlungen aus dem mathematischen seminar der hamburg gavril intersection graphs subtrees trees exactly chordal graphs journal combinatorial theory golumbic algorithmic graph theory perfect graphs elsevier science edition edition dan gusfield perfect phylogeny problem missing removable data solutions via chordal graph theory journal computational biology gysel potential maximal clique algorithms perfect phylogeny problems arxiv gysel gusfield extensions improvements chordal graph approach multistate perfect phylogeny problem transactions computational biology bioinformatics gysel lam gusfield constructing perfect phylogenies proper triangulations characters algorithms molecular biology hara takemura boundary cliques clique trees perfect sequences maximal cliques chordal graph technical report metr department mathematical informatics university tokyo heggernes minimal triangulations graphs survey discrete mathematics lee counting clique trees computing perfect elimination schemes parallel information processing letters kloks kratsch spinrad treewidth minimum asteroidal graphs theoretical computer science kumar madhavan clique tree generalization new subclasses chordal graphs discrete applied mathematics lin mckee west leafage chordal graph discussiones mathematicae graph theory mckee mcmorris topics intersection graph theory number siam monographs discrete mathematics applications meacham theoretical computational considerations compatibility qualitative taxonomic characters felsenstein editor numerical taxonomy volume nato asi series pages parra scheffler characterizations algorithmic applications chordal graph embeddings discrete applied mathematics rose triangulated graphs elimination process journal mathematical analysis applications semple steel characterization set partial partitions define discrete mathematics semple steel phylogenetics oxford lecture series mathematics applications oxford university press steel complexity reconstructing trees qualitative characters subtrees journal classification walter representations chordal graphs subtrees tree journal graph theory
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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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generalized value iteration networks life beyond lattices sufeng siheng hanyu colin melissa jelena oct clemson university calhoun clemson usa carnegie mellon university forbes avenue pittsburgh usa abstract paper introduce generalized value iteration network gvin neural network planning module gvin emulates value iteration algorithm using novel graph convolution operator enables gvin learn plan irregular spatial graphs propose three novel differentiable kernels graph convolution operators show kernel achieves best performance furthermore present episodic improvement upon traditional stabilizes training vin gvin lastly evaluate gvin planning problems mazes irregular graphs realworld street networks showing gvin generalizes well arbitrary graphs unseen graphs larger scale outperforms naive generalization vin discretizing spatial graph image introduction reinforcement learning technique solves sequential decision making problems lacks explicit rules labels sutton barto recent developments deep reinforcement learning drl lead enormous progress autonomous driving bojarski innovation robot control levine humanlevel performance atari games mnih guo board game silver given reinforcement learning task agent explores underlying markov decision process mdp bellman bertsekas attempts learn mapping state space data optimal policy maximizes expected return reinforcement learning categorized lillicrap mnih approaches sutton barto deisenroth rasmussen schmidhuber approaches learn policy directly attempt avoid bias caused suboptimal environment model sutton barto majority recent architectures drl follow approach lillicrap mnih approaches hand allow agent explicitly learn mechanisms environment lead strong generalization abilities equal contribution copyright association advancement artificial intelligence rights reserved recent work value iteration networks vin tamar combines recurrent convolutional neural networks emulate process value iteration bellman bertsekas vin learns environment plan shortest paths unseen mazes input data fed deep learning systems usually associated regular structures example speech signals natural language underlying sequential structure images underlying lattice structure take advantage regularly structured data deep learning uses series basic operations defined regular domain convolution uniform pooling however data contained regular structures urban science traffic information associated road networks neuroscience brain activity associated brain connectivity networks social sciences users profile information associated social networks learn data irregular structure recent works extended lattice structure general graphs defferrard bresson vandergheynst kipf welling redefined convolution pooling operations graphs however works evaluate data fixed given graph addition lack ability generalize new unseen environments paper aim enable agent plan optimal path new unseen spatial graphs using drl techniques task relevant many applications route planning cars web proposed method general classical drl extending irregular structures furthermore proposed method scalable computational complexity proportional number edges testing graph handles various edge weight settings adaptively learns environment model note optimal path necessarily shortest one additionally proposed work differs conventional planning algorithms example dijkstra algorithm requires known model gvin aims learn general model via trial error apply said model new unseen irregular graphs create gvin generalize vin two aspects first work irregular graphs propose graph convolution operator generalizes original convolution operator new graph convolution operator proposed network captures basic concepts spatial graphs direction distance edge weight also able transfer knowledge learned one graph others second improve reinforcement learning irregular graphs propose reinforcement learning algorithm episodic stabilizes training vin gvin original vin trained either imitation learning requires large number labels reinforcement learning whose performance relatively poor proposed episodic new network performs significantly better vin reinforcement learning mode since proposed network generalizes original vin model call generalized value iteration network gvin main contributions paper proposed architecture gvin generalizes vin tamar handle regular structures irregular structures gvin offers architecture trained via reinforcement learning labels see section framework proposed graph convolution operator generalizes convolution learns concepts direction distance enables gvin transfer knowledge one graph another see section graph convolution proposed reinforcement learning algorithm episodic extends classical monte carlo control significantly improves performance reinforcement learning irregular graphs see section training via reinforcement learning intensive experiments demonstrate generalization ability gvin within imitation learning episodic various datasets including synthetic maze data irregular graphs maps minnesota highway new york street maps show gvin significantly outperforms vin discretization input irregular structures see section experimental results background markov decision process consider environment defined mdp contains set states set actions reward function series transition probabilities probability moving current state next state given action goal mdp find policy maximizes expected return accumulated rewards immediate reward time stamp discount rate policy probability taking action state value state policy expected return starting following value taking action state policy expected return starting taking action following least one policy better equal policies called optimal policy optimal policy arg optimal function optimal function obtain usually consider solving bellman equation value iteration popular algorithm used solve bellman equation discrete pstate space iteratively compute maxa convergence differentiable planning module vin employs embedded differentiable planning architecture trained via imitation learning tamar vin bellman equation encoded within convolutional neural networks policy obtained backpropagation however vin limited regular lattices requires imitation learning maximum performance trained separately reactive policy recent work memory augmented control network macn khan combines vin model memory augmented controller backtrack history previous trajectories however shown later table gvin outperform macn performance problem scales different work predictron uses learning planning model simulates markov reward process silver architecture unrolls imagined plan via predictron core however predictron limited markov rewards process relatively computationally expensive compared vin deep learning graphs number recent works consider using neural networks handle signals supported graphs niepert ahmed kutzkov duvenaud henaff bruna lecun principal idea generalize basic operations regular domain filtering pooling graph domain based spectral graph theory example bruna henaff bruna lecun introduce hierarchical clustering graphs spectrum graph laplacian neural networks defferrard bresson vandergheynst generalizes classical convolutional neural networks using graph coarsening localized convolutional graph filtering kipf welling considers learning graphs using convolutional neural networks investigate learning graph structure gated recurrent unit gilmer considers message passing framework unifies previous work see recent overviews bronstein methodology propose new drl framework gvin takes general graph starting node goal node inputs outputs designed plan goal gvin learn underlying mdp summarizes optimal planning policy applied arbitrary graphs requires gvin capture general knowledge planning structure transition invariant depend specific graph structure key component mdp transition matrix needed solve bellman equation train general transition matrix works arbitrary graphs similar vin treat graph convolution operator parameterize using graphbased kernel functions represents unique action pattern train parameters gvin using episodic makes reinforcement learning irregular graphs practical figure architecture gvin left module emulates value iteration obtains state values right module responsible selecting action based policy training greed policy testing emphasize contributions including graph convolution operator episodic blue blocks framework input gvin graph starting node goal node training phase gvin trains parameters various graphs testing phase gvin plans optimal path based trained parameters framework includes planning module left action module right shown figure planning module emulates value iteration iteratively operating graph convolution action module takes greedy action according value function mathematically consider directed weighted spatial graph node set node embeddings ith row embedding ith node consider spatial graphs method generalizable edge set adjacency matrix element representing edge weight ith jth nodes consider graph signal mapping nodes real values use graph signal encode goal node activates goal node let reward graph signal graph signal graph signal respectively represent entire process matrixvector form follows max step encoded become robust reward via function convolutional neural network case regular graphs identity function operating irregular graphs step graph convolution operator ath channel set graph convolution operators trained based graph described section graph convolution value iteration emulated using graph convolution obtain graph signal ath channel obtain graph signal training parameters parameterize respectively shown figure repeat graph convolution operation iterations obtain final graph signal lattice planning module gvin degenerates vin training phase feed final graph action module original vin extracts signal action values step trains final action probabilities eight directions however problematic irregular graphs number actions neighbors node varies solve consider converting whose sth pseudo graph signal representing action element value moving one neighbors advantages approach come following three aspects final state value node obtained using maximum action values across channels robust small variations pseudo graph signal considers unique action node depend number actions node agent queries state values neighbors always moves one highest value pseudo graph signal considers local graph structure next state always chosen one neighbors current state pseudo graph signal used episodic learns experience backpropagates update training parameters episodic episode obtained follows given starting node agent move sequentially strategy probability arg probability randomly selected one neighbors episode terminates goal state maximum step threshold reached episode consider loss function bst bst function training parameters gvin episode length expected return time stamp defined discount factor immediate return time stamp additional details algorithm discussed section training via reinforcement learning testing phase obtain action greedily selecting maximal state value arg graph convolution conventional cnn takes image input lattice graph node pixel local structure sitting grid connecting eight neighbors case convolution operator easy obtain irregular graphs however nodes form diverse local structures making challenging obtain structured translation invariant operator transfers knowledge one graph another fundamental problem find convolution operator works arbitrary local structures solve learning spatial kernel function provides transition probability distribution space according evaluate weight edge obtain graph convolution operator spatial kernel function assigns value position space reflects possibility transit corresponding position mathematically transition probability starting position another position spatial kernel function specified later definition spatial kernel function shift invariant satisfies shift invariance requires transition probability depend relative position key transfer learning words matter starting position transition probability distribution invariant based spatial kernel function graph adjacency matrix obtain graph convolution operator element kernel function kwp parameterized embeddings ith jth node graph convolution operator follows graph connectivity spatial kernel function property spatial kernel function leads local transition distribution node graph adjacency matrix works modulator select activations graph convolution operator edge edge high kwp high words transition probability ith node jth node higher edge weight high influence ith node jth node bigger graph convolution note sparse matrix sparsity pattern corresponding adjacency matrix ensures cheap computation shown graph convolution multiplication graph convolution operator graph signal see figure note work lattice graph appropriate kernel function graph convolution operator nothing matrix representation conventional convolution lecun bengio others words vin special case gvin underlying graph lattice see details supplementary kernel functions consider three types spatial kernel functions directional kernel spatial kernel embedding kernel figure multiplication graph convolution graph convolution operator diffuses graph obtain graph signal directional kernel directional kernel embedded direction information element graph convolution operator models probability following edge cos kernel coefficient direction edge connecting ith thejth nodes computed node embeddings directional kernel order reference direction reflecting center activation hyperparameters include number directional kernels order reflecting directional resolution larger indicates focus one direction see figure kernel coefficient reference direction training parameters spatial kernel next consider direction distance element graph convolution operator dij cos dij distance ith jth nodes computed node embeddings spatial kernel reference distance reference direction indicator function otherwise hyperparameters include number directional kernels order reference distance distance threshold kernel coefficient reference direction training parameters kernel directional kernel spatial kernel manually design kernel provide hints gvin learn useful patterns directly feed node embeddings allow gvin automatically learn implicit hidden factors general planning element graph convolution operator indicator function otherwise kernel function kemb mnnet mnnet standard neural network training parameters weights neural network practice graph weighted may also include graph adjacency matrix input multilayer neural network theorem proposed three kernel functions directional kernel spatial kernel kernel shift invariant proof follows fact kernels use direction distance difference two node embeddings depend relative position training via reinforcement learning train gvin episodic modified version difference episodic nstep fixed episode duration updates training weights steps episodic episodic terminates agent reaches goal maximum step threshold reached update trainable weights entire episode experiments found regular irregular graphs policy planned original keeps changing converge due frequent updates similar monte carlo algorithms sutton barto episodic first selects actions using exploration policy goal reached afterwards accumulate gradients entire episode update trainable weights allowing agent use stable plan complete entire episode simple change greatly improves performance see section revisting mazes pseudocode algorithm presented algorithm supplementary episodic experimental results section evaluate proposed method three types graphs mazes synthesized irregular graphs real road networks first validate proposed gvin comparable original vin mazes regular lattice structure next show proposed gvin automatically learns concepts direction distance synthesized irregular graphs reinforcement learning setting without using labels finally use gvin model plan paths minnesota road network manhattan street network additional experiment parameter settings listed supplementary experiment settings revisting mazes given starting point goal location consider planning shortest paths mazes see figure supplementary example generate mazes using vin used use configuration vin data training data testing consider four comparisons vin gvin based imitating learning statevalue based imitating learning gvin unguided gvin reinforcement learning four metrics used quantify planning performance including prediction probability taking action state higher means better success probability successfully arriving goal start state without hitting obstacles higher means better path average length difference predicted path path lower means better expected average accumulated reward higher means better overall testing results summarized table vin gvin gvin performs competitively vin table especially gvin uses actionvalue based imitation learning column table outperforms others four metrics figure supplementary shows value map learned gvin based imitation learning see negative values blue obstacles positive values red around goal similar value map vin reported tamar vin imitation learning slightly outperforms vin imitation learning similarly gvin based imitation learning slightly outperforms gvin based imitation learning results suggest action approximation method section framework impact performance maintaining ability extended irregular graphs gvin unaware gvin directionaware gvin slightly outperforms gvin reasonable fixed eight directions ground truth regular mazes remains encouraging gvin able find directions imitation learning shown later gvin outperforms gvin irregular graphs figures show planning performance improves kernel exponential increases due resolution reference direction low small figure supplementary compares kernel reference direction two different kernel orders kernel activates directions kernel focuses directions higher resolution https vin prediction accuracy success rate path difference expected reward gvin unaware unaware table maze performance comparison vin gvin gvin achieves similar performance vin mazes imitation learning achieves similar performance imitation learning prediction accuracy mazes success rate mazes prediction accuracy irregular graphs success rate irregular graphs figure kernel direction order influences planning performance regular irregular graphs based kernel scale generalization reinforcement learning imitation learning use performance metrics previously discussed maze experiments expected rewards success rate figure episodic maze reinforcement learning also examine performance episodic section training via reinforcement learning vin table supplementary shows episodic algorithm outperforms training method used vin trpo curriculum learning results reported table able train vin using algorithm episodic epochs trpo curriculum learning took epochs train vin reported tamar algorithms used settings shown figure episodic algorithm shows faster convergence better overall performance compared exploring irregular graphs consider four comparisons following experiments directional kernel spatial kernel directional kernel spatial kernel embeddingbased kernel first train gvin via imitation learning table shows kernel outperforms kernel methods terms action prediction path difference column table indicating kernel captures edge weight information distance within neural network weights better methods spatial kernel demonstrates higher accuracy success rate compared directional kernel suggests effectiveness using bin sampling method shows slightly better results spatial kernel larger success rate gain directional kernel figure supplementary shows visualization learned value map shares similar properties regular graph value map also train vin column converting graph image shown table vin fails significantly see supplementary experiment settings figures show planning performance irregular domain kernel order increases results show larger irregular domain opposite effect compared regular domain observation reasonable irregular domain direction neighbor extremely variable larger kernel order creates narrower direction range seen prediction acc success rate path diff expected reward vin macn nodes directional kernel unaware spatial kernel unaware kernel train train train table performance comparison amongst vin three different kernels gvin experiments except macn khan tested irregular graphs note last column trained using episodic stands imitate learning reinforcement learning respectively similar experimental settings macn achieves success rate graphs gvin achieves success rate graphs details training vin irregular graphs sees section irregular graphs supplementary material prediction accuracy success rate path difference expected reward optimal minnesota optimal new york city table performance comparison minnesota new york city street map data using gvin trained graphs trained graphs expected rewards success rate figure episodic irregular graphs thus resulting information loss reinforcement learning train gvin using episodic compare imitation learning baseline also train gvin using standard deep qlearning techniques including using experience replay buffer target network networks use kernel function kernel configurations figure shows comparison two algorithms success rate expected rewards training clearly episodic converges high success rate high expected rewards standard deep techniques fail achieve reasonable results scale generalization also examine scale generalization training graphs testing graphs using kernel gvin trained graphs via imitation learning performance significantly hindered shown table column gvin trained using episodic table column shows excellent generalization abilities outperform imitation learning based results success rate expected rewards compared imitation learning also observe performance decreases path differences action prediction graph edge weights also test gvin handles edge weights set true weighted shortest path wij distance two nodes wij edge weight shown table imitation learning trained graphs reinforcement learning trained also examine gvin excluding edge weights input see effects performance table supplementary shows reinforcement learning edge weights slightly help agent find suitable policy imitation learning input edge weights cause significant failure validating real road networks demonstrate generalization capabilities gvin evaluate two maps minnesota highway map contains nodes representing intersections edges representing roads new york city street map contains nodes representing intersections edges representing roads use models trained graphs containing nodes kernel using episodic section exploring irregular graphs separately normalize data coordinates set recurrence parameter randomly pick start points goal points different times use algorithm baseline table shows generalize well large scale data policy could reach goal position experiments one sample planned path shown supplementary figures conclusions introduced gvin differentiable novel planning module capable regular irregular graph navigation impressive scale generalization also introduced episodic designed stabilize training process vin gvin proposed graph convolution may applied many applications navigation point cloud processing molecular analysis left future works references bellman dynamic programming princeton usa princeton university press bertsekas bertsekas bertsekas bertsekas dynamic programming optimal control volume athena scientific belmont bojarski del testa dworakowski firner flepp goyal jackel monfort muller zhang end end learning cars arxiv preprint bresenham linear algorithm incremental digital display circular arcs communications acm bronstein bruna lecun szlam vandergheynst geometric deep learning going beyond euclidean data arxiv preprint bruna zaremba szlam lecun spectral networks locally connected networks graphs arxiv preprint defferrard bresson vandergheynst convolutional neural networks graphs fast localized spectral filtering advances neural information processing systems deisenroth rasmussen pilco modelbased approach policy search proceedings international conference machine learning duvenaud maclaurin iparraguirre bombarell hirzel adams convolutional networks graphs learning molecular fingerprints advances neural information processing systems gilmer schoenholz riley vinyals dahl neural message passing quantum chemistry arxiv preprint guo singh lee lewis wang deep learning atari game play using offline tree search planning advances neural information processing systems henaff bruna lecun deep convolutional networks data arxiv preprint khan zhang atanasov karydis kumar lee memory augmented control networks arxiv preprint kipf welling classification graph convolutional networks arxiv preprint lecun bengio convolutional networks images speech time series levine finn darrell abbeel training deep visuomotor policies arxiv preprint tarlow brockschmidt zemel gated graph sequence neural networks arxiv preprint lillicrap hunt pritzel heess erez tassa silver wierstra continuous control deep reinforcement learning arxiv preprint mnih kavukcuoglu silver graves antonoglou wierstra riedmiller playing atari deep reinforcement learning arxiv preprint mnih badia mirza graves lillicrap harley silver kavukcuoglu asynchronous methods deep reinforcement learning international conference machine learning niepert ahmed kutzkov learning convolutional neural networks graphs proceedings annual international conference machine learning acm schmidhuber algorithm dynamic reinforcement learning planning reactive environments neural networks ijcnn international joint conference ieee silver huang maddison guez sifre van den driessche schrittwieser antonoglou panneershelvam lanctot mastering game deep neural networks tree search nature silver van hasselt hessel schaul guez harley reichert rabinowitz barreto predictron learning planning arxiv preprint sutton barto reinforcement learning introduction volume mit press cambridge tamar levine abbeel thomas value iteration networks advances neural information processing systems tieleman hinton lecture divide gradient running average recent magnitude coursera neural networks machine learning appendix computational complexity let input graph nodes edges testing phase since input graph commonly sparse computational complexities based sparse computation respectively therefore total computational complexity number iterations spatial graph number edges usually proportional number nodes thus computational complexity scalable huge graphs episodic highlight differences episodic original blue including initial expected return termination condition timing updating gradient algorithm episodic input graph goal initialize global step counter initialize gvin parameters initialize parameter gradients repeat one episode clear gradients randomly pick start node repeat one action take action according policy based qst receive reward new state terminal tmax accumulate gradients wrt end tmax kernel functions directional kernel first consider direction face several roads intersection straightforward pick one whose direction points goal aim use directional kernel capture edge direction parameterize graph convolution operation element graph convolution operator models probability following edge cos kernel coefficient direction edge connecting ith thejth nodes computed node embeddings directional kernel order reference direction reflecting center activation hyperparameters include number directional kernels order reflecting directional resolution larger indicates focus one direction see figure kernel coefficient reference direction training parameters note graph convolution operator sparse matrix sparsity pattern input adjacency matrix ensures computation cheap figure directional kernel function activates areas around reference direction spatial domain activated area concentrated increases intuition behind graph convolution operator represents unique direction pattern edge vector sampled spatial plane direction edge connecting matches one reference directions higher probability follow edge gvin consider several channels channel obtain action value node represents matching coefficient direction pattern operation selects matching direction pattern node spatial kernel next consider direction distance roads current intersection opposite goal straightforward try shortest one first thus include edge length consideration element graph convolution operator dij cos dij distance ith jth nodes computed node embeddings spatial kernel reference distance reference direction indicator function otherwise hyperparameters include number directional kernels order reference distance distance threshold kernel coefficient reference direction training parameters figure spatial kernel function activates areas around reference direction reference distance spatial domain compared directional kernel spatial kernel adds another dimension distance words directional kernel special case spatial kernel ignore distance spatial kernel activates localized area plane spatial kernel graph convolution operator represents unique directiondistance pattern edge connecting matches one reference directions distances higher probability follow edge kernel directional kernel spatial kernel manually design kernel hint gvin learn useful patterns directly feed node embeddings allow gvin automatically learn implicit hidden factors general planning element graph convolution operator indicator function otherwise kernel function kemb mnnet aij mnnet standard neural network training parameters weights neural network note graph convolution operator still sparse matrix sparsity pattern input adjacency matrix plus identity matrix directional kernel spatial kernel implicitly discretize space based reference direction distance input pair given direction distance kernel function outputs response based closed reference direction distance embeddingpi success rate epochs path difference expected reward trpo table performance comparison using different training algorithms vin model first column vin trained trpo curriculum learning reported tamar second column vin trained episodic qlearning based kernel set reference direction distance discretize space instead use neural network directly regress arbitrary edge edge weight embedding representation response value kernel thus flexible directional kernel spatial kernel may learn hidden factors experiment settings implementation based tensorflow gpuenabled platform experiments use standard centered rmsprop algorithm optimizer learning rate tieleman hinton reinforcement learning experiments use discount rmsprop decay factor exploration rate annealed linearly first epochs mazes experiments set follows consider rules follows agent receives reward reaching goal receives reward hitting obstacle movement gets reward preprocess input data use cnn vin gvin first layer involves kernels size second layer involves kernel size output transition probability matrix parameterized convolution kernels size vin gvin gvin use directional kernel based method shown equations set represent eight reference directions consider two approaches initialize directions directionaware approach fix approach set weights train via backpropagation set recurrence gvin mazes regular domain set kernel order default irregular graphs evaluate proposed methods section graph convolution irregular domain experimental domain synthetic data consists irregular graphs graph contains nodes follow standard rules random geometric graphs generate irregular graphs specifically generate vertices random coordinates box connects pair two vertices edge distance two vertices smaller certain threshold node graph define input map mazes value map mazes input map irregular graph value map irregular graph figure value map visualization regular irregular graph imitation learning edge weight edge weight prediction accuracy success rate path difference expected reward reinforcement learning edge weight edge weight table performance comparison testing weighted graphs imitation learning trained irregular graphs reinforcement learning trained irregular graphs nates represent spatial position ranged dataset split graphs training graphs testing additionally exam whether gvin could handle weighted graphs also generated synthetic dataset consisting node graphs partitioned graphs training graphs testing directional kernel spatial kernel set number reference directions kernel order default values equations also set interval mode set trainable weights mode spatial kernel function set number bins equation kernel use three layers fully connected neural networks layer uses relu max activation function neural network initialized derivation three kernel methods set graph convolution channel number recurrence training vin irregular graphs show strong generalization gvin evaluate vin irregular graph converting graph data format image testing set contains reward map obstacle map sizes pixels use weights maze parameters tuned highest performance make training testing consistent set rewards map obstacle map settings maze vertices edges marked free path value set area marked obstacles value set edge path generated via bresenham line algorithm bresenham set recurrence value iteration could cover whole map gvin prediction figure sample planning trajectories minnesota highway map gvin prediction figure sample planning trajectories new york city street map
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theory application control systems linearly parametrizable uncertainty bound aug spandan roy sayan basu roy indra narayan kar senior member ieee work proposes new control arc architecture class uncertain systems upper bound uncertainty satisfies linear parameters lip structure conventional arc strategies either require structural knowledge system presume overall uncertainties time derivative norm bounded constant due unmodelled dynamics modelling imperfection true structural knowledge system always available class systems consideration prior assumption regarding uncertainties time derivative upper bounded constant puts restriction states beforehand conventional arc laws invite problem switching gain towards front adaptive based robust control asrc proposed alleviates overestimationunderestimation problem switching gain moreover asrc avoids presumption constant upper bound overall uncertainties negotiate uncertainties regardless linear nonlinear parameters experimental results asrc using wheeled mobile robot notes improved control performance comparison adaptive sliding mode control index control systems wheeled mobile robot uncertainty ntroduction background controller design aspect nonlinear systems subjected parametric nonparametric uncertainties always challenging task adaptive control robust control two popular control strategies deal uncertain nonlinear systems case adaptive control online computation unknown system parameters controller gains complex systems significantly intensive front robust control reduces computation burden complex systems compared adaptive control requiring predefined upper bound uncertainties however practice always possible estimate prior uncertainty bound due effect unmodelled dynamics increase operating region controller often higher uncertainty bounds assumed turn leads overestimation switching gain high control effort considering individual constraints adaptive robust control recently global research reoriented towards control arc series publications roy basu roy kar department electrical engineering indian institute new delhi india sayanetce ink regarding arc estimate individual uncertain system parameters adaptive law robust control utilized negate effect external disturbances works utilize projection operator respective adaptive laws necessitate knowledge lower upper bound individual uncertain system parameters adaptive sliding mode control asmc designed parameter identification mechanical servo systems lugre friction considering uncertainties linear parameters lip contrast controllers assume overall uncertainty time derivative bounded constant thereafter constant term estimated adaptive law rather estimating individual uncertain system parameters adaptive laws involve predefined threshold value matter fact threshold value achieved switching gain may still increasing resp decreasing even tracking error decreases resp increases thus creates overestimation resp underestimation problem switching underestimation problem compromises controller accuracy applying lower switching gain required amount overestimation problem causes larger gain high control input requirement adaptive law reported requires predefined bound time derivative uncertainties observed method also requires frequency characteristics perturbation design filter equivalent control however work assumed time derivative uncertainties bounded unknown constant motivation let consider following system representing chemostat operating monod kinetic states uncertain positive parameters known constant control input system following relations hold inspection uncertainties reveal though nonlinear parameters nlip upper bounds lip similarly systems uncertainties lip nlip system nonlinear friction structure however upper bound overall lumped uncertainty systems lip property systems general represent large class real world systems like robotic manipulators mobile robots ship dynamics aircraft pneumatic muscles etc systems immense applications various domains defence automation industry surveillance space missions etc controllers assume overall uncertainties upper bounded constant assume time derivative overall uncertainty bounded constant hence aforementioned class systems consideration constant bound known unknown restricts system state priori switching gain suffers underestimation problems practice also always possible prior knowledge bounds system parameters required projection operator denotes system state denotes vector generalized control input forces represents matrix denotes coriolis centripetal terms denotes gravity vector represents vector slip damping friction forces denotes bounded external disturbances system possesses following properties property matrix skew symmetric property property matrix uniformly positive definite exist two positive constants property let desired trajectory tracked selected let tracking error filtered tracking error contribution view discussion importance systems scenarios imperative formulate dedicated arc framework uncertain systems towards front adaptive based robust control asrc presented paper tracking control uncertain systems formulation asrc insensitive towards nature uncertainties negotiate uncertainties either lip nlip asrc utilizes lip structure upper bound uncertainty presume overall uncertainty time derivative upper bounded constant adaptive law asrc prevents switching gain becoming monotonically increasing function allowing switching gain decrease within finite time tracking error decreases moreover asrc alleviates problem switching gain realize effectiveness performance asrc compared asmc experimentally using pioneer wheeled mobile robot wmr organization notations remainder article organized follows proposed asrc framework second order systems detailed section followed experimental results asrc comparison asmc section iii section presents concluding remarks following notations used paper represent minimum eigenvalue euclidean norm respectively denotes identity matrix appropriate dimension denotes set positive real numbers positive definite matrix multiplying time derivative using yields represents overall uncertainty implies characterization upper bound relation system property yields system properties provide following since verified using upper bound holds following lip structure problem formulation general system second order dynamics written let following condition always holds ontroller esign robust controller system designed gef provides robustness switching gain small scalar used chattering removal positive definite matrix evaluation switching gain evaluation like conservative nature evidently requires knowledge always possible face uncertain parametric variations external disturbances control laws developed assume upper bounded constant respectively exploring structure easily inferred constant bound assumption uncertainties whether known unknwon puts restriction states priori moreover switching gain suffers problem adaptive based robust control asrc major aims proposed asrc framework compensate uncertainties either lip nlip however upper bound uncertainties satisfies lip property alleviate problem switching gain control input proposed asrc designed gef provides robustness estimate auxiliary gain importance explained later gains evaluated using following adaptive laws otherwise otherwise remark tracking error remains bounded inside ball using relation implies switching gains sufficient enough keep error within hence gains kept unchanged one choose small improve tracking accuracy gets reduced long value invite chattering remark initial condition gains selected adaptive laws force gains increase either gains attempt breach respective lower bounds governed ensures gains remain unchanged hence combination conditions mentioned implies condition later exploited stability analysis guarantee alleviation overestimation problem switching gain necessary decrease within finite time shown theorem theorem let tin time instant gains start increasing exist finite times gains decrease tin max times obtained etf min max proof tin time gains start increasing solely used analysis objective theorem find gains start decrease noted laws gains increase sufficient investigate condition gains increase moreover using one proposed law certainly make switching gain monotonically increasing function thus alleviates overestimation problem initial time user defined scalars substituting closed loop system formed gef cef remark noticed adaptive laws gains increase error trajectories move away governed decrease error trajectories move away governed otherwise condition implies hence first laws condition yields let lypaunov function using relation time derivative yields etf etf etf gef etf substituting using property implying etf simplified stability analysis exploring structures adaptive laws three possible scenarios identified case increase gef etf case decrease case gef theorem closed loop system control input guarantees uniformly ultimately bounded uub thus sufficient condition achieve would proof stability analysis overall system carried three cases mentioned using following common lyapunov function let system posses finite time escape integrating sides inequalities using results lead expressions tin case increase note using following procedure one obtains min definition yields tin tin gains increasing ensure otherwise condition condition increase needs take place definition upper bound follows let tin implies tin decreases exponentially tin following exist finite time tin tin implying tin would occur tin would start decreasing time found tin tin gef max substituting using comparison lemma yields tin gef etf inferred thus definition yields thus case system remains stable gains remain bounded case decrease case remain constant case hence case decrease using yields gef etf gef since definition yields remark increment decrement occur several times depending error incurred system however time interval two successive decrement always satisfy moreover high values help reduce achieve faster adaptation thus using substitution yields applying descarte rule sign change one verify maximum one positive real root noticed hence according bolzano intermediate value theorem least one positive real root combination intermediate value theorem descarte rule sign change reveals exactly one positive real root therefore nature roots either one positive real root three negative real roots one positive real root one negative real root pair complex conjugate roots let positive real root figure depicts nature depending various combination roots noted actual graph values roots polynomial would depend values coefficients however unknown nevertheless study stability system sufficient analyse nature instances rather determining values roots nature polynomial understood occurrence real roots moreover leading coefficient coefficient highest degree term negative matter fact one notice fig hence overall system would uub case point view controller design important reduce better tracking accuracy achieved obtained increasing similar case simplified case gef etf gef system remains bounded inside ball implies hence let define scalar using modified min hence system would uub since closed loop system remains uub cases using common lyapunov function overall closed loop system also remains uub remark noteworthy condition necessary stability system moreover high values helps reduce consequently improve controller accuracy however one needs careful high value may excite condition leading increment gains scalar terms used purpose analysis used design control law remark importance auxiliary gain realized theorems observed gets reduced due presence contributed leads faster adaptation moreover negative fourth degree term contributed ensures system stability case making also indicates reason selecting lower bounds gains selected zero figure one positive real root three negative real roots one positive real root one negative real root pair complex conjugate roots hence overall system would uub case case special case quadratic term uncertainty bound contributed matrix property systems robotic manipulator underwater vehicles ship dynamics etc includes however also exist second order system reduced order wmr system term systems following lip structure would hold hence following switching gain laws control laws uncertainty structure modified otherwise otherwise insufficient error increase however increase rather keeps decreasing creates underestimation problem low resp high value may force increase resp decrease longer duration resp resulting escalation overestimation resp underestimation problem asmc whereas asrc allows gains decrease error trajectories move towards overcoming overestimation problem keeps gains unchanged sufficient keep error within ball overcoming underestimation problem since problems alleviated asrc one fact reduce better tracking accuracy long chattering occur iii pplication onholonomic wmr system stability employing analysed exactly like theorem using following lyapunov function one verify cubic polynomial case would modified quadratic polynomial using hence following argument remark noticed cubic term selected adaptive law system stability thus system two structures possible situations covered better inference asrc algorithm summarized table various system structures table asrc lgorithm various ystem tructures system structure lip structure control law comparison existing law gain insight advantage proposed adaptive law following adaptive law asmc switching gain provided user defined scalars sliding surface observed switching gain increases monotonically even error trajectories move close gives rise overestimation problem switching gain even sufficient keep within decreases monotonically certain time would become figure schematic wmr nonholonomic wmr vast applications transportation planetary exploration surveillance security provides unique platform test proposed control law hence performance proposed asrc verified using commercially available pioneer wmr comparison adaptive sliding mode control asmc asmc law detailed follows adaptive law formulation nonholonomic wmr fig given generalized coordinate vector system coordinates center mass system heading angle rotation torque inputs right represent left wheels respectively system mass system inertia wheel inertia wheel radius robot width respectively distance center line joining two wheel axis represent constraint matrix vector constraint forces lagrange multipliers respectively expressions found noteworthy system two control inputs although five generalized coordinates fact wmr one directly control wheel positions rather system dynamics represented combination reduced order dynamics kinematic model efficient controller design specifically face unmodeled dynamics timevarying uncertainty experiment payload system may varied due addition removal sensors according application requirement causes variations overall system mass center mass inertia etc original systems dynamics formed based pure rolling assumption assumption satisfied practice due friction effect wheel surface denoted wmr dynamics apart also effects external disturbances however evaluation switching gain robust controller like requires prior knowledge bound uncertainties implies designer needs knowledge wmr due absence coriolis term wmr dynamics means designer knowledge parametric variations systems well bound demands tedious modelling job also always accurate benefit applicability efficacy proposed asrc realized context asrc require knowledge systems dynamics terms wmr system matter fact sysb cos dsin cos dsin tem representing dynamics implementing sin dcos sin dcos control law need knowledge rather adapts terms adaptive law since coriolis component zero asrc algorithm applied wmr based control laws hence asrc eliminates effort model system well avoids need characterize uncertain parameters disturbances noted used coordinate transformation wmr pose representation control law design hence objective apply asrc asmc reduced order wmr system track desired effect track desired illustrate fact one direct wmr move specified circular path designing two suitable different fixed wheel velocities path applying approximated wmr moves ground gravity vector velocity profile wheels potential function would certainly zero implies satisfies properties main implication system property hold etf experimental scenario easily verified wmr wmr directed follow circular path using dynamics based rolling without slipping condition following desired trajectories hence term omitted however practical rad rad circumstances wmr always subjected uncertainties like friction slip skid external disturbance etc hence incorpopioneer uses two quadrature incremental encoders rating system dynamics modified ppr always starts initial wheel position error rad helps realize error convergence ability controllers desired considered unmodelled wmr pose xdc ycd actual wmr pose namics disturbance respectively often simple controllers determined using open loop control olc pid controller used obtained encoder respectively practice simplicity however various works supplied manufacturer references reference control laws asrc asmc written discussed need formulate advanced robust environment considering hardware response time tracking controllers wmr compared conventional open sampling interval selected controllers loop control pid control improve tracking accuracy create dynamic payload variation payload added kept sec removed sec periodically robotic platform different places controller parameters asrc selected controller parameters asmc selected experimental results comparison path tracking performance asrc depicted fig following desired circular path tracking performance comparison asrc asmc illustrated fig terms defined euclidean distance error defined asmc framework built assumption uncertainties upper bounded unknown constant general systems particular wmr based experiment assumption restrictive nature systems switching gain thus insufficient provide necessary robustness matter fact asrc provides better tracking accuracy asmc evaluate benefit proposed law evaluation switching gain asmc asrc provided fig respectively figure reveals switching gain asmc increases even approaches towards time sec due fact decrease unless gives rise overestimation problem hand asrc seen fig gains decreases decrease sec asrc overcomes overestimation problem encountered asmc decreases monotonically time durations sec monotonic decrement makes insufficient tackle uncertainties certain time creating underestimation problem result starts increasing sec leading poor tracking accuracy increases gains asrc contrary stay unchanged sec gains sufficient keep avoiding underestimation problem evaluation asrc shown fig noted wmr hence though remain constant sec constant due presence shown magnifying time durations sec sec fig reduction would cause overestimation problem asmc table shows tracking accuracy asrc improves reduced control parameters kept unchanged chattering observed control input figure performance comparison asrc asmc figure switching gain evaluation asmc table erformance asrc various rms root mean squared rms figure circular path tracking asrc noticed fig initials gains high enough decreases beginning gains hence would prudent verify capability asrc alleviating figure performance comparison various controllers figure switching gain evaluation asrc figure switching gain evaluation gains figure evaluation asrc underestimation problem starting relatively low gains therefore experiment asrc repeated much lower initial value gains previously initial values time selected tracking performance evaluation switching gain later case shown fig respectively denotes denotes noticed fig initially tracking error high compared asmc initial gain due low initial gains however tracking accuracy begins improve gains became sufficient enough negotiate uncertainties eventually tracking performance found similar much improved compared asmc proves proposed adaptive law perform satisfactorily even low initial conditions another important aspect verify whether alleviate issue similar verified fig gains follow pattern according adaptive laws due initial low values increases gains similarly gains decrease decreases gains remain unchanged sufficient keep filtered tracking error according law thus overcoming underestimation problem moreover gains increase decrease sand thus avoids overestimation problem hence low initial gain conditions affect capability alleviating problem moreover noticed first conditions rate increment hence similar value fig increases please note selected experiment however rate falling different second conditions thus exhibit different falling pattern fig onclusions novel asrc law proposed class uncertain systems upper bound uncertainty possesses lip structure benefit asrc lies fact independent nature uncertainty negotiate uncertainties lip nlip asrc presume overall uncertainty bounded constant avoids putting prior restriction states moreover proposed adaptive law alleviates overestimationunderestimation problem switching gain experimental results validate efficacy proposed control law comparison existing adaptive sliding mode control future work would extend asrc law systems unmatched disturbances eferences liu yao chu adaptive robust control class uncertain nonlinear systems unknown sinusoidal disturbances ieee conference decision control bandyopadhayay janardhanan spurgeon advances sliding mode control new york zhu tao yao cao adaptive robust posture control parallel manipulator driven pneumatic muscles redundancy transactions mechatronics vol zhu tao yao cao integrated adaptive robust posture control parallel manipulator driven pneumatic muscles ieee transactions control system technology vol zhang chen lee adaptive robust control servo mechanisms partially known states via dynamic surface control approach ieee transactions control system technology vol pan nonlinear adaptive robust control actuator unknown nonlinear parameters ieee transactions control system technology vol yao chen wang adaptive robust repetitive control industrial biaxial precision gantry contouring tasks ieee transactions control system technology vol sun zhao gao saturated adaptive robust control active suspension systems ieee transactions industrial electronics vol islam liu saddik robust control four rotor unmanned aerial vehicle disturbance uncertainty ieee transactions industrial electronics vol chen yao wang based adaptive robust posture control linear motor driven stages high frequency dynamics case study transactions mechatronics vol liu pan new adaptive sliding mode control uncertain nonlinear dynamics asian journal control vol chen tao nan ren adaptive nonlinear sliding mode control mechanical servo system lugre friction compensation asme journal dynamic systems measurement control vol doi nasiri nguang swain adaptive sliding mode control class mimo nonlinear systems uncertainty journal franklin institute vol meng zhang gao song adaptive sliding mode control uncertain stewart platform based offline multibody dynamics transactions mechatronics vol liu zhou luo xiao adaptive sliding fault tolerant control nonlinear uncertain active suspension systems journal franklin institute doi mobayen adaptive pid sliding mode control based dynamic sliding manifolds class uncertain nonlinear systems nonlinear dynamics doi plestan shtessel bregeault poznyak new methodologies adaptive sliding mode control international journal control vol plestan shtessel bregeault poznyak sliding mode control gain adaptation application electropneumatic actuator state control engineering practice vol utkin poznyak adaptive sliding mode control application algorithm equivalent control method automatica vol moreno negrete fridman adaptive continuous twisting algorithm international journal control doi adetola guay lehrer adaptive estimation class nonlinearly parametrized dynamical systems ieee transactions automatic control vol annaswamy skantze loh adaptive control continuous time systems parametrization automatica vol spong hutchinson vidyasagar robot dynamics control john wiley sons new york shang cong motion control parallel manipulators using acceleration feedback ieee transactions control system technology vol choi choi seo position compliance control pneumatic muscle actuated manipulator enhanced safety ieee transactions control system technology vol roy nandy ray shome time delay sliding mode control nonholonomic wheeled mobile robot experimental validation ieee international conference robotics automation huang wang xing nonlinear disturbance observerbased dynamic surface control trajectory tracking pneumatic muscle system ieee transactions control system technology vol haddad chellaboina nonlinear dynamical systems control approach princeton university press khalil nonlinear systems prentice hall das kar design implementation adaptive fuzzy controller wheeled mobile robots ieee transactions control system technology vol anderson jackson sitharam descartes rule signs revisited american mathematical monthly vol russ translation bolzano paper intermediate value theorem historia mathematica vol leung mok suen polynomials functions honk kong university press liberzon switching systems control coelho nunes control mobile robots presence uncertainties ieee transactions robotics automation vol campion bastin modelling state feedback control nonholonomic mechanical systems ieee conference decision control
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fast make interpreted python russell power alex rubinsteyn aug new york university power alexr abstract python popular dynamic language large part appeal coming powerful libraries extension modules augment language make productive environment wide variety tasks ranging web development django numerical analysis numpy unfortunately python performance quite poor compared modern implementations languages lua javascript python lag far behind languages show api extension libraries make python powerful language also make difficult efficiently execute given want retain access great extension libraries already exist python fast make evaluate designed implemented falcon bytecode interpreter fully compatible standard cpython interpreter falcon applies number well known optimizations introduces several new techniques speed execution python bytecode evaluation found falcon average faster standard python interpreter benchmarks cases faster introduction python popular programming language long history active development community major driver python popularity diverse ecosystem libraries extension modules make easy almost anything writing numerical computing despite significant effort python developers performance python interpreter still lags far behind implementations languages lua javascript differentiates python faster languages obviously implementation choices jit interpreter dramatic effect performance case python however landscape choices severely constrained api makes easy extend standard interpreter python called cpython exposes api python api allows building extension libraries embedding interpreter programs python api allows access almost every aspect interpreter including inspecting current interpreter state threads running function stacks etc pulling apart representation various object types integer float list dictionary performance reasons many python libraries written another compiled language interface python via api ideal world implementation python semiformal specification would interchangeable cpython implementation unfortunately avoid breaking libraries alternative implementation must also support full api size api functions burdensome really makes problematic degree exposes internal memory layout behavior python objects result many python extensions become intimately coupled current implementation cpython interpreter instance modifying layout basic object format example use less memory breaks even source level compatibility existing extensions value extensions python hard overstate python already fast jit compiler form pypy seen widespread adoption large extent due lack support existing cpython extension libraries replacing libraries simple undertaking numpy alone consists almost lines source scipy libraries build upon another lines evaluate much improve cpython without breaking existing extension modules developed alternative python virtual machine called falcon falcon converts cpython bytecode format performs optimizations remove unnecessary operations occasionally elide type checks falcon register bytecode executed threaded interpreter attempts accelerate common code patterns using attribute lookup caching register tagging overall found combination register conversion simple bytecode optimization virtual machine threaded dispatch result average speedup cpython certain benchmarks falcon faster array local values essence register bytecode every instruction explicitly labels registers local variable slots reads register writes result translated register code example would look like register conversion optimization fast enough run online every function means system like falcon accelerate code without need profiling compilation heuristics figure register code adding two inputs converted register code local variables represented registers since source destination registers part instruction possible express function using two instructions downside falcon register code format like register code instruction must larger make room register arguments two advantages register code make space increase worthwhile first potential reducing time spent virtual machine dispatch reducing number instructions must executed previous research verified switching virtual machine register machines improve performance additionally much easier write optimizations register code reason every instruction implicitly affects stack program analyses optimizations must track side effects form virtual stack register code hand admits expression compact optimization implementations section since instructions depend explicit flows data along named registers overview falcon replace standard cpython interpreter rather runs inside function marked execution falcon decorator falcon decorated function called falcon translates function stack bytecode falcon compact register code register code optimized remove redundant computations decrease number registers needed function optimized register code passed falcon virtual machine evaluation example consider executing following function adds two inputs assigns sum local variable returns variable def add return figure python function adds two inputs python first encounters code compiled following bytecode compiler figure python stack code adds two inputs operation python bytecode implicitly interacts value stack python first pushes values local variables onto stack instruction takes two values adds pushes new result value onto stack values local variables come addition stack python virtual machine maintains distinct array values named local variables numerical arguments attached instructions indicate local variable loaded stored even simple example see python bytecode burdens virtual machine great deal wasteful stack manipulations could get better performance away stack instead used falcon compiler structured series passes modifies register code way illustrate behavior pass use simple example function called counts number elements list given threshold def return sum python generates following stack bytecode figure anything else need convert original stack machine bytecode equivalent register code conversion convert python stack code register code falcon uses abstract interpretation falcon takes form virtual stack stores register names instead values falcon steps function stack operations evaluates effect virtual stack emit equivalent register machine operation sum python falcon stack instructions directly alias variables specially designated register names simplifies code reduces number instructions needed python falcon stack figure python stack machine bytecode handling control flow code process fairly easy python instructions fairly straightforward effect stack happens encounter branch need properly simulate execution paths handle situation must make copy virtual stack evaluate sides branch branches come merge points places two branches execution come together thread control flow might assigned different register names stack position handle situation falcon inserts rename instructions merge points ensuring incoming register stacks compatible mechanism employed compilers use static single assignment form ssa resolve register aliased local variable therefore operation need generate falcon instruction instead simply push onto virtual stack pops sequence stack pushes back iterator sequence python falcon stack branch instruction either pushes next element iterator onto stack next instruction pops iterator stack jumps side loop python falcon stack example conversion let walk works example stack code figure first find value function sum using instruction cpython interpreter looks particular name dictionary global values pushes value onto stack since set literal names used function known compile time instruction simply reference index string sum table constant names equivalent register machine instruction assigns global value fresh register case brevity stack column listings show register number instruction python falcon stack effect operation virtual stack push register top later operation consumes inputs stack correctly wired use argument constructs empty list contain results create new register push onto stack python special operations load store local variables load constants rather implement one branch instruction takes inner loop continues iterator exhausted python falcon append stack behavior instruction might look somewhat surprising appears peek stack find special behavior unique instruction likely result past performance tuning cpython interpreter building lists common operation python branch takes function epilogue python falcon sum sum stack operations dynamic stack effects example effect instruction stack known statically turns case almost python instructions fact one operation stack effect must determined runtime instruction appears functions try finally block determines whether caught exception possible handle instruction dynamically inserting branches generated code possible stack effect chose much simpler option simply compile functions containing instruction instead functions evaluated using existing python interpreter instruction relatively rare occurring functions python standard library almost never found performance sensitive code cost supporting minimal stead use original source code figure copy propagation changes optimization improve performance instead enables optimizations remove useless instructions reuse unoccupied registers dead code elimination value contained register never used program likely occur copy propagation may possible delete instruction created register instructions lack side effects simple moves registers safe delete whereas instructions may run code must preserved even result goes unused copy propagation applied code register never used thus move instruction gets deleted dead code elimination bytecode optimizations stack register pass bytecode looks like figure sum figure unoptimized register code note rather using positions code jump targets falcon splits code basic blocks indicated prefix change effect code ultimately gets run virtual machine greatly simplifies implementation optimization passes register machine bytecode emitted pass tends uses many registers often contains redundant loads used emulate effect stack operations improve performance perform number optimizations remove redundant operations improve register usage advantage switching register code becomes clear point apply known optimization techniques unoptimized register code almost modification optimizations used falcon common compilers briefly describe application falcon copy propagation whenever value copied registers possible change later uses target register register renaming even register used point program might necessarily alive entire duration function execution two registers nonoverlapping live ranges may possible keep one replace uses register reduces total number registers needed run function saving memory giving slight performance boost register code optimization falcon optimizations applied bytecode extraneous store instructions removed furthermore register renaming causes registers used repeatedly place several registers optimized register bytecode achieves greater instruction density compared original stack code optimization general reduces number instructions similar improvement performance sum figure optimized register code difficulty optimizing python code would desirable run even compiler optimizations invariant code motion common subexpression elimination unfortunately valid applied python bytecode reason optimizations invalid almost python operation might trigger execution code unrestricted side effects example might tempting treat second following example redundant however due possibility encountering overloaded method assumptions made behavior general absence type information almost every instruction must treated virtual machine compilation register code passed virtual machine evaluate falcon uses main techniques lookup hints try improve dispatch performance cover section common straightforward approach used python writing bytecode interpreter use switch loop code switch opcode case add reg reg reg break case sub reg reg reg break figure switch dispatch compilers generate efficient based dispatch switch statement problem style dispatch make effective use branch prediction unit cpu since every instruction dispatched top switch statement cpu unable effectively determine instructions tend follow others leads pipeline stalls poor performance fortunately easy way improve technique improving performance switch based interpreters basic idea inline behavior switch statement end every instruction requires compiler supports labels values available compilers microsoft compiler notable exception inlining jump table lookup replace single difficult predict branch many predictable add reg reg reg goto opcode sub reg reg reg goto opcode figure token threading branches example always followed certain loop processor able accurately predict branch avoid stalling token threading recently added python interpreter one step modify bytecode contain actual address handler instruction results foreach instr function instr handler opcode add reg reg reg goto handler figure direct threading direct threading increases size instruction removing lookup jump table implemented token direct threading falcon token threading seems provide modest performance improvement switch based dispatch tagged registers default python object format pyobject inefficient simple integer requires words space must dereferenced get actual value numerically intensive code cost indirection dominate interpreter runtime overhead reduced using efficient object format general changing object format would break compatibility existing code falcon use registers proves convenient long value register store whatever format efficient value handed python api external call need convert back normal python object format tests chose simple tagged integer format integer tagging takes advantage fact object pointers always aligned memory word boundaries least significant bits always zero therefore use least significant bit register indicate whether storing integer value pointer storing integer shift register right one bit obtain value boland provides detailed descriptions different tagged representations simplified example tagged registers implemented show figure struct register union int pyobject bool return int return mask bottom bits pyobject return figure tagged register format lookup hints attribute lookups handled instruction account significant portion time spent python interpreter compilers many languages accelerate method lookups using polymorphic inline caching pic shadow classes many instances replace expensive dictionary direct pointer offsets making effectively costly struct unfortunately technique difficult employ context falcon reasons nately would increase complexity falcon order magnitude may appropriate choice future interested determining whether simpler approach might effective falcon uses variant pic call lookup hints name suggests provide guess attribute found hint records location attribute found last time instruction run hint indicate attribute instance dictionary object attribute parent class hint found location specified hint checked first normal lookup traversal performed hint matches resulting value returned immediately otherwise full lookup procedure performed new hint generated benefit hints depends greatly application executed code references lots attributes consistent manner see large improvement performance general observed improvement benchmarks void loadattr regop eval load hint eval hint pyobject reg pyobject reg pyobject key reg found instance dictionary look dict size klass keys offset key return values offset hint normal path fixed object format shadow classes require control objects laid memory would break compatibility goal complex lookup behavior python provides great deal flexibility application programmers choosing attribute lookup performed builtin behavior resolving lookups similar found languages first check object dictionary class dictionary parent classes addition python offers unusual degree flexibility programmers form accessor methods allow specifying happens attribute found using method even completely override normal lookup process method methods added class objects created creates complications mechanisms like expect lookup behavior remain consistent attribute lookup hidden bytecode making matters worse bytecode generated python explicitly check whether methods defined instead instruction expected implicitly perform checks various accessor functions every lookup one general way handling complex behavior use traces within compiler figure simplified implementation lookup hints implementation falcon implemented python extension module compatible python building top existing python interpreter means falcon takes relatively small amount code implement entire package lines code evenly split interpreter method used implement falcon somewhat unusual typical interpreter structured single method containing code handle type instruction approach took falcon implemented number classes one python opcode use compiler function attributes force inlining code opcode main interpreter dispatch loop found technique effective allowing clean separation code without sacrificing speed like cpython interpreter falcon overlays python function calls onto execution stack python call corresponds function call python exceptions emulated using exceptions allows falcon leverage builtin exception handling facility greatly simplifies main interpreter code proper handling exceptions return values significant source complexity mainline python interpreter compile times negligible benchmark sults include time taken compile stack register code run optimization passes despite best efforts making inefficient compiler copies used many places references might suffice multiple passes code time taken convert optimize fan tre thr fas cry pto ult rdc fan tre thr fas cry pto rdc ult figure shows runtime performance falcon relative runtime cpython interpeter three bars represent time taken unoptimized falcon code using untagged ordinary pyobject registers optimized code untagged registers optimized code tagged registers surprised inconsistency benefit using tagged registers benchmarks matrix multiplication performance improvement switching tagged registers quite dramatic benchmarks saw either little improvement even slowdown switch also looked change number instructions used converting register code optimizations run figure expected register code version benchmark requires significantly fewer instructions express computation using average fewer instructions interesting observations made ick ick figure benchmark descriptions unoptimized optimized description multiply square matrices recursive control flow distinct words aes classic sorting algorithm random string generation values threshold count permutations figure falcon performance relative python evaluated runtime performance falcon variety benchmarks tests performed machine memory xeon processor benchmarks falcon provides small performance benefit cpython interpreter benchmarks bound loop interpreter dispatch overhead falcon twice fast cpython evaluation benchmark matrix multiplication decision tree wordcount crypto quicksort fasta count threshold fannkuch unoptimized optimized figure effect compiler optimizations number opcodes relative number python stack operations functions small varying simple functions complex function benchmark set aes encryption implies profitable simply convert everything register code rather relying techniques used psyco determine whether worthwhile optimization important benchmarks compiler optimizations result improvement unoptimized code cases changing falcon slower cpython significantly faster register code amenable optimization register machine instructions slower expensive dispatch simple stack machine operations bit tagging registers yields mixed results switching compact efficient internal representation seemed like would straightforward win always case potential benefit using tagged inline integer value must weighed potential cost converting integers python objects whenever must passed api functions functions dominated arithmetic logic operations tagged registers performance win functions however unpacking integer value stored directly register simply wasted work related work many different projects sought speed performance python programs using variety techniques nuitka cython shedskin reduce runtime overhead opcode dispatch statically compiling python programs api calls approach combined aggressive optimization also remove redundant runtime checks disadvantage approach requires explicit sometimes lengthy compilation step odds usual programming style dynamic language like python currently popular approach accelerating dynamic language tracing jit compilation proven particularly effective javascript one primary ways jit able achieve good performance using unboxed representations data incompatible native api exposes internal representation data unfortunately case python currently active jit project python pypy although pypy able achieve impressive performance gains expense breaking api compatibility particularly problematic scientific python libraries act largely wrappers fortran code often written particular expectations python object layout psyco older abandoned compiler python coupling intimately python interpreter switching efficient unboxed representations externally compatible boxed representations psyco able avoid breaking extensions unfortunately compatibility required great deal conceptual implementation complexity eventually drove developers abandon project favor pypy conclusion investigate fast could make binary compatible interpreter python built falcon fast register based compiler virtual machine python falcon combines many techniques new ones order achieve significant speedup regular python learn experience stack register bytecodes different register based interpreter proved faster basic python stack interpreter tasks nice improvement much larger gains could made ability change object format tagged object formats important performance improvement using inline tagged format integer nan tagging primitive types worth extra effort sort performance sensitive code easily means difference interpreter times slower one times slower type object format could used uniformly within cpython interpreter would greatly improve performance almost every task api design important lesson draw experience interpreter apis designed care particular api exposes internals interpreter work may convenient gaining quick performance boost use macro instead function exposing internal surfaces makes nearly impossible change improve performance future python size api problem rather insistence particular object format assumption made api always obvious instance writing api interpreter may tempting functions directly take return object pointers simple decision unexpected consequences prevents use copying garbage collector future work one goals falcon build platform would simplify writing new experiments use register code pass based compiler format makes trying new optimization techniques python bytecode easy particular ideas would like explore future include type specialization compile time type propagation performed determine types registers unboxed type specific bytecode generated leverage information container specialization performance benefit tagged registers primarily limited need convert cpython object format whenever api call made almost always due register stored python list dictionary object improve creating specialized versions lists dictionaries primitive type specialized objects would support standard interface convert python object format demand thus allowing used external code internally would store objects efficient tagged format improving attribute hints current falcon hinting mechanism improves performance slightly limited application better results could obtained making lookup explicit bytecode first check accessor functions look actual name ranz type specialization dynamic languages pldi source code falcon available online http encourage anyone interested try provide feedback hambers ngar optimizing languages polymorphic inline caches ecoop european conference programming springer references liphant python scientific computing computing science engineering cython python http gcc documentation labels values accessed august python token threading accessed august api reference manual accessed august nderson ettig performing lisp analysis fannkuch benchmark bala uesterwald banerjia dynamo transparent dynamic optimization system pldi vol acm ell threaded code commun acm june oland memory allocation access patterns dynamic languages phd thesis heinrich heine university dsseldorf olz uni ijalkowski igo tracing pypy tracing jit compiler proceedings workshop implementation compilation optimization languages programming systems acm ousot ousot abstract interpretation unified lattice model static analysis programs construction approximation fixpoints proceedings acm symposium principles programming languages acm ytron errante rosen egman adeck efficiently computing static single assignment form control dependence graph acm transactions programming languages systems oct davis eatty asey regg waldron case virtual register machines interpreters virtual machines emulators ivme acm press ebenita ranz one method time quite waste time proceedings second ecoop workshop implementation compilation optimization languages programs systems ich haver nderson delin aghighat aplan oare barsky rendorff ruderman mith eitmaier ebenita hang igo specialization psyco prototype python proceedings acm sigplan symposium partial evaluation program manipulation acm asey rtl regg virtual machine showdown stack versus registers acm trans archit code optim van rossum python programming language
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logic programming introductory computer science course high school students timothy maritz yuanlin university texas san antonio usa university texas austin usa texas tech university usa abstract paper investigates high school students approach computing introductory computer science course situated logic programming paradigm study shows novice students operate within paradigm engaging foundational computing concepts skills presents case viable paradigm choice introductory courses keywords education high school declarative programming logic programming answer set programming introduction debate paradigm selection type courses often split procedural paradigms also leads discussions programming language choice selection difficult task standard languages paradigms use field though approaches successful introductory courses research also shown minimal differences comparing outcomes paradigms time research teaching introductory courses also reveals limitations approaches suitability java education debatable well however logic programming based approach largely ignored community despite continuing efforts teaching research logic programming education prolog may instance used teach particular prolog used teach children pioneer kowalski focused declarative aspects prolog restricted use procedural aspects prolog minimum later researchers practitioners found procedural aspects prolog main source misconceptions difficulties benefits declarative aspects acknowledged last two decades prolog made appearances listed high school curriculum taught gifted talented high school students general high students though enjoy attention procedural paradigms advantages logic programming existing work practice general programming main advantages related computing education simple syntax semantics natural connection abstraction logic reasoning knowledge representation form foundation computing disciplines involvement interesting challenging problems sudoku problem early mathematical flavor also provides rich context embedded essential computing concepts skills industrial relevance declarative programming general seen profound application impact database query languages problem formal specification languages domain specification languages including popular web application development languages html css xslt etc breakthrough research last two decades establishment answer set programming asp paradigm inherited declarative nature prolog fully getting rid procedural features currently asp dominating formalism knowledge representation research questions although movements expanding research asp full declarative nature facilitates student learning introductory computer science still absent research question study adopting asp teach introductory course high school students impact understanding computer science computing methods study adopts qualitative approach exploring high students understanding computing specifically asp setting participants participants recruited texprep program texas tech university summer texprep summer program graders offers full day courses science engineering math computer science several weeks courses typically taught local grade teachers well university faculty members students case study class taught third author tas second author also texprep selective program students average letters recommendations written personal interview consideration prospective participants fourth year texprep taking computer science already taken scratch alice courses sequence texprep course teaches foundations computing using asp course emphasized problem solving knowledge representation asp paradigm course met monday friday informed consent obtained participants parents prior start class sixteen participants study seven females nine males participants students high school texas tech area lubbock terms rising grades eleven graders three graders two graders data collection procedures surveys participants responded surveys questions beginning pre end post course regarding experiences computing declarative programming purpose surveys provide background information participants knowledge triangulate findings clinical interviews survey refers declarative programming rather asp even though study focused latter two paradigms rationale referring general asp belongs paradigm one paradigm knowing paradigms functional programming may give edge students learning asp asked participants general knowledge computer science declarative programming questions computer science describe know declarative programming asked participants knowledge computer science topics course questions specifically asked participants learned course understood concepts rather asking participants directly explain declarative programming questions asked perceived declarative programming computer science learned anything taken course feel understood subject course tasks asked clinical interviews clinical interviews activities researcher attempts explore subjects understanding cognitive process active questioning probing last week class participants asked work one problem thinking aloud since class assignment limited time researchers participants worked part assignment clinical interviews conducted research team participant researchers asked probing questions order prompt participants explain clarify thought processes researchers received training first third author conducting clinical interviews clinical interviews lasted minutes participant completed came first clinical interviews video recorded transcribed interview task student asked write sparc program represent knowledge family relationships relations father denoting father mother given program stub including declarations relations also given students asked represent following knowledge jon father matthew following relations grandparent son aunt descendant sparc programming language system instance asp paradigm offers type system overcome challenging syntax restrictions existing asp systems dlv clingo help discover programming errors early results surveys interviews presented following sections survey results survey responses transcribed analyzed researchers using inductive coding approach main unit analysis sentence longer sentences representing multiple ideas sometimes separated coding similarly consecutive sentences grouped together analysis continued line thought analysis researchers found common themes across participants responses grouped common themes presented participants given pseudonyms presentation results form students know computer science since course third program sequence participants already two courses computing science engineering courses participants definition computer science centered creation program task programming language means tasks completed science behind computer functions performs tasks computer science using different computer languages get computers perform tasks understanding able reproduce code programs make computer task strong connection computer science programming ten participants mentioning programming creation software responses participants acknowledged importance computers technologies advancement society study develop computers help world easier efficient place become extremely prominent daily lives must always improved technology come end course participants asked question three participants mentioned programming definition computer science participants responses could categorized defining computer science problem solving computer science study involves mathematics helps understand way computer thinks computer science study make computer like human get computers solve problems use simplifying numbers solutions problems study technologies science behind computer functions computer science study computers work thus end course participants definition broadened bit away programming rather become holistic emphasizing problem solving order complete tasks students know declarative programming participants know however many familiar traditional procedural programming prior programming experience eight participants said know declarative programming participants offered explanation declarative programming could would guess branch computer science making statements receiving form output statements believe give description object program runs find item similar described explanation done programming school learned little java java declarative language know much else declarative programming explanations participants attempting make connections previous programming experiences even though incorrect remaining five participants explicitly say know rather offered incorrect incomplete definitions declarative programming stating statement able program statement giving computer command programming accomplish particular command program things type declarative programming elaboration algorithmic programming developed broader use compared algorithmic programming much elaborates well suited situations similarities type programming needs defined output action achieved participant closest correct answer know declarative programming works like telling computer rules something computer rules something computer find somethings sic set thus participants experience though experience paradigms included procedural imperative students learned course data showed students reported learning program sparc language within context lab assignments though explicitly articulate understanding languages prior experiences able report solved problems using asp methodologies tools although participants emphasized learning program sparc eleven responses implied programming language provided structure required students think problems deeper thorough perspective lends declarative programming response representative learning programming language helped guide declarative understanding learned reevaluate thinking process consider basis thought program learned translate writing english transferring modifying fit sparc code similar responses include learned look problem different way learned sort objects define relationships predicates use multiple variables use multiple rules use tex rules variables use rules help simplify solutions learned new coding language along others know valuable also learned new way look relations objects new syntax new way specify relations examples show rules structure syntax semantics sparc required associated methodologies helped guide participants problem solving four participants emphasized learning syntax coding style also mentioned first course type programming rather interface students understood course topics tasks eight participants stated understood subject course within confines course activities three indifferent understood learn respect programming language think given time work computer could use language future prompt things asked clear easy understand easy pick gave general understanding making relatively easy complete given task understand subject easy pretty sure wanted program complex things would harder two participants stated able understand subject course unsure applicable real world everyday setting understand understand purpose type program beneficial world artificial intelligence feel though program requires get point feel understood subject course actually learn something however always understand exactly supposed occasionally completely lost therefore participants knew learn beyond contexts presented two participants stated understand course topics mainly due time management course felt course repetitive simple topics challenging enough similarly felt course also repetitive instructor teaching topics already found notes interview results grounded theory methodology adopted generate theory explained students understood approached computing asp activities based interview data qualitative approach generates theory explanation systematic analysis data grounded theory goes beyond describing categorizing finding processes relationships phenomenon studied approach conducted clinical interview data served main analysis study unit analysis utterance consisted complete thought phrases spoken participant cases utterances rich many potential units broken smaller utterances utterances made researcher provided context made participant analyzed utterances used analysis first author analyzed data process coding utterance descriptive label process required several iterations ensure coding consistent across entire dataset third author coded data using set labels created first author process researchers debated labels coding utterances led several iterations recoding data revision initial labels memos kept analysis process regarding rationale behind labels formation coding since third author expert field asp taught course coding used remaining analysis next step creating categories represented larger constructs happening data axial coding analyzed way categories related based properties formed larger categories case study many merged labels found relate well infrequent labels five major categories abstraction representation reasoning revision procedural first author led first stage axial coding led five major categories also debated discussed researchers surveys particularly used guide axial coding creation categories definitions categories also went several revisions next sections define category give example quotes abstraction category describes instances participants understand problem space general abstract terms thinking higher level real world terms within context code writing participants generally describing relationships respect terms integrating prior knowledge problem space example quotes included descendant part descendants anything person well case father person person parent sister parent gender thinking like family tree guess sort thinking thinking thinking person whose descendants looking representation category refers process participants trying understand problem space coding terms abstraction understanding problem translates code category includes references conceptual coding well transition process two example quotes included say parent son flipped hope explaining right saying son parent putting gender son male male yes right going move aunt descendant like child umm like child two parents descendant heir trying say know say put son put predicates son person know example put matthew son reasoning category represents problem solving process describes instances participants actively thinking problem apply strategies associated asp adapting existing code solve current problem reasoning selected category name frequent related label example quotes included grandchild isaac george grandchild joseph george grandchild susie george grandchild children grandchild grandchild grandchild grandfather grandfather define gender yes gender male grandfather male like separate ones like sister like aunt parent nephew isaac actually create rule revision revision encapsulates lot overall debugging process emphasizes importance participants asking questions sparc see code correct category includes participants process questioning see completed tasks correctly done querying see solutions right example quotes included write aunt write rose comma susie add question mark press execute need write queries see program works going ask descendants george george like patriarch procedural category represents large part data participants either restating interpreting facts instructions cases participants asking correct interpretation assignment instructions category participants frequently mentioned specific rules strategies associated declarative programing often utterances descriptive categories representation reasoning example quotes included whenever talking son forgot include male male son forgot add periods got got got need aunt jon father matthew rule wrote works order category occurrences part axial coding analysis categories occurred clinical interview learn categories related categories graph plotted occurrence category order appear participant plots varied length due variance number coded utterances figure shows example graph utterance one category associated plots stacked fig sample order category occurrence visual analysis participants graphs conducted describe category occurred relationships occurrences across participants interviewers read instructions beginning utterances coded since participants asked work assignments participants already started prior interview thus graphs immediately started labels visual analysis found following revision ten participants engaged revision last half interview expected spent first half identifying objects relationships knowledge coding testing get far interview engaged revision participants already started assignments interview quicker revisions abstraction evidence abstraction mostly came beginning middle participants several participants occurred alongside procedural participants well representation participants three participants abstraction revision may suggest questioning process may support part abstraction abilities another three participants showed evidence abstraction interview however interviewer reported get chance test program technical issues laptop reported feel confident started revision early reasoning reasoning came throughout interview participants similar procedural utterances may suggest important recurring process reasoning occurred instances revision representation abstraction thus reasoning also intertwined parts asp process evidence reasoning however many coded utterances participant general representation similar reasoning procedural representation also happened throughout interview part representation started clusters revision participants however three participants revision afterward representation intertwined revision throughout interview interview consisted utterances labeled representation mostly listing represented knowledge base procedural participants instances procedural utterances distributed throughout interview well simultaneously codes interesting observation procedural moments throughout interview beginning participants kept going back instructions facts presented assignment although procedural category seem provide much insight participants approached computing computer science first utterances participants reading rules asking understood instructions information correctly procedural utterances made throughout entire overall computing process selective coding analysis examines category could connected categories procedural found central theme identifying objects relations problem english definitions carefully reading problem description integrating one common sense knowledge major component explicit methodology solving problems thus deep interaction procedural processes students also mentioned post surveys learned methodology solve problems although may surprising showed able adopt strategies turn led reasoning representation processes model participants approach based grounded theory analysis supported visual analysis ordering categories model constructed may explain participants approach computing based data see figure figure highlights founfig model participants approach dational importance following strategies declarative programming case answer set programming provided explicit methodology guided students towards abstracting concepts representing code reasoning problem solving abstraction representation reasoning often happened together grouped separately revision mostly came toward end task similarly abstraction representation happened sequentially abstraction coming representation reasoning observed throughout interview indicated larger rectangle model showed importance procedures case students able follow explicit methodology students also demonstrated general problem solving skill iterative refinement two distinguishable big loop components knowledge specification revision knowledge specification iterative refinement using interleaving steps abstraction representation reasoning example students try specify knowledge aunt try figure objects individual persons relations whose sister whose parent try define knowledge aunt using objects relations participants tried make sure definition captures intended meaning requires reason understand relations better fact model reflects intended skills researchers would like students obtain introductory course skills general important computing beyond skills include procedure general methodology problem solving iterative refinement problem solving programming revision category name abstraction rigorous high level logic representation knowledge logic based reasoning knowledge thought introducing explicit problem solving methodology problem understanding precise representation reasoning knowledge methodology almost universal problem solving also particularly useful lays foundation problem understanding knowledge representation discussions students done well lab assignments final interview tasks together pre post surveys shows easy learn main reason simple syntax semantics asp allow students focus less language specific feature problem solving skills clinical interview data shows students able learn apply intensively important concepts underlying computing model figure shows able apply explicit methodologies problem solving iterative refinement big steps knowledge specification revision small steps abstraction representation reasoning solving problems participants applied abstraction representation reasoning heavily interwoven manner understanding extracting defining knowledge needed solving given problem knowledge needed able carry standard programming tasks labeled revision categories coding debugging program model generated showed students able engage computing skills abstraction problem solving representation debugging revision indeed able explain computing concepts within context course tasks students able operate within paradigm driven explicit methodology however may immediately see applicable outside context assignments understood concepts processes successful completing assignments work needs done make connections see real world students understood problem solving nature believed one way solving problems limitations participants average students program experience scratch alice however participants stated experience though every student initially volunteered study still participants stopped attending complete assignment lastly course met four weeks provided limited exposure compared regular university courses future studies introductory courses could conducted full semester courses rather short courses conclusions paper asserts educators take closer look using introductory courses suggested future computer scientists equipped foundational programming language principles involving logic formal specification design implement complex software systems needed society results show students able focus key concepts computing including abstraction representation reasoning solving problems simple syntax intuitive semantics asp allows put less attention language specific features findings support viable option teach introductory course using asp acknowledgments authors acknowledge cynthia perez rocky upchurch contributions project thank michael gelfond sharing teaching materials work partially supported nsf grant references draft computer science curricula chen monge simon relationship early programming language novice generated design acm sigcse bulletin vol vilner zur fundamental concepts procedural object oriented case study acm sigcse bulletin pears seidman malmi mannila adams bennedsen survey literature teaching introductory programming acm sigcse bulletin vol chakravarty keller risks benefits teaching purely functional programming first year journal functional programming vol kowalski logic computer language children nichol dean briggs prolog children students nichols publishing mendelsohn green brna programming languages education search easy start psychology programming scherz haberman logic programming based curriculum high school students use abstract data types acm sigcse bulletin stutterheim swierstra swierstra forty hours declarative programming teaching prolog junior college utrecht arxiv preprint beux briola corradi delzanno ferrando frassetto computational thinking beginners successful experience using prolog ball zorn teach foundational language principles communications acm vol kowalski logic programming computational logic vol siekmann elsevier gelfond kahl knowledge representation reasoning design intelligent agents programming approach cambridge university press dovier benoli brocato dereani tabacco reasoning high schools asp italian conference computational logic strauss corbin basics qualitative research techniques procedures developing grounded theory thousand oaks sage publications mit media lab scratch carnegie mellon university alice ginsburg entering child mind clinical interview psychological research practice new york cambridge university press balai gelfond zhang towards answer set programming sorts international conference logic programming nonmonotonic reasoning alviano faber leone perri pfeifer terracina disjunctive datalog system dlv datalog reloaded springer gebser kaufmann kaminski ostrowski schaub schneider potassco potsdam answer set solving collection communications vol
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aug row graded betti table toric variety alexander lemmens abstract prove explicit formula first entry row graded betti table projective toric variety associated normal polytope least one interior lattice point applies veronese embeddings also prove explicit formula entire row interior polytope onedimensional results valid arbitrary field index syzygies toric varieties lattice polytopes koszul cohomology contents introduction toric varieties graded betti tables combinatorial proof introduction let field article study syzygies projectively embedded toric varieties precisely give explicit formulas terms combinatorics defining polytope certain graded betti numbers appear minimal free resolution homogeneous coordinate ring graded module obtained repeatedly taking syzygies betti numbers typically gathered graded betti table number degree summands module resolution one alternatively defines dimension koszul final publication http available springer via cohomology space graded betti table expected contain wealth geometric information subject several important open problems conjectures vast part poorly understood number entries explicit formula terms defining lattice polytope known examples found paper relevant result schenck proved projective toric surfaces coming lattice polygon lattice boundary points hering proved theorem using theorem theorem next entry zero results already known case polygon equal triangle vertices polygon gives veronese embedding projective plane loose proved number zeroes quadratic strand equals counting zero result independently rediscovered ottaviani paoletti generalized following conjecture conjecture veronese embedding projective space whenever known property generalized following conjecture conjecture authors already proved conjecture take minimal free resolution line bundle opn veronese embedding degree syzygies veronese embeddings still active area research short introduction syzygies toric varieties refer reader next section prove conjecture prove explicit formula first entry row also prove formula first entry row betti table projectively normal toric variety dimension row zero note row last row zero work arbitrary field convex lattice polytope denote convex hull lattice points topological interior ready formulate main result theorem let toric variety coming normal polytope let interior polytope let number translations contained number lines parallel disjoint first statement actually follows green linear syzygy theorem theorem combined koszul duality second statement already appeared conjectural formula information koszul cohomology toric surfaces found recall whenever note whenever follows proposition theorem context conjecture first entry row equals two theorems proved end section using results section corollary toric surfaces coming polygons lattice width two know entire betti table explicitly max area course everything else zero second formula comes lemma first follows directly theorem using theorem one deduce following formula graded betti table canonical model tetragonal curve toric surface max tetragonal invariants introduced schreyer genus actually formula true tetragonal curves follows explicit minimal free graded resolution schreyer article include explicit formula easy find literature section explain toric varieties syzygies koszul cohomology prove theorems using results section use koszul duality expresses betti numbers row terms betti numbers first row betti table serre dual line bundle core article section construct explicit basis last entry first row graded betti table graded module form line bunldes normal projective toric variety comes constructing basis kernel map dim theorems proved results section namely theorem actually also follows green linear syzygy theorem theorem corollary theorem acknowledgements article part thesis funded research foundation flanders fwo colleagues wouter castryck filip cools noticed patterns betti tables certain toric surfaces motivated find explicit basis also grateful referee carefully reading article making many useful suggestions also want thank milena hering bringing article attention toric varieties graded betti tables projectively normal toric varieties work arbitrary field lattice points mean points projective toric varieties built polytopes convex hull finite set lattice points works follows suppose ndimensional let list lattice points define embedding coordinate let closure image happens positive integers polytope called normal case projective toric variety corresponding projectively normal example veronese embedding projective space given polytope following form veronese embedding instance get embedding monomials correspond lattice points triangle vertices taking zariski closure image corresponds standard veronese embedding normal one still take integer multiples normal sufficiently large one associate toric variety large enough normal variety depend embedding however simplicity restrict case normal homogeneous coordinate ring given mean vector space spanned monomials possibly negative exponents xinn corresponding lattice points mean ample line bundle coming embedding projective space graded betti tables given projective variety homogeneous coordinate ring consider graded tor modules toris note graded module symmetric algebra graded tor modules computed either taking graded free resolution syzygies taking graded free resolution koszul cohomology mainly work latter graded betti table table nonnegative integers column row defined dimension degree part torps general table following shape example convex hull minimal graded free resolution vector space spanned monomials corresponding lattice points symmetric algebra polynomial ring variables homogeneous coordinate ring image corresponds ideal cutting veronese surface ideal generated six elements constitute minimal set generators ideal free graded module rank six polynomial ring basis elements degree two mapped generators ideal makes sure morphism modules means image consists relations generators called syzygies turns minimal generating set eight syzygies degree rank graded free module basis elements degree turns rank basis elements degree gives graded betti table note polynomial ring free module rank one monomial generator koszul cohomology let line bundles complete variety let symmetric algebra graded module define koszul cohomology space homology following sequence indicates removed wedge product write denote dimension resp resp ample line bundle coming projective embedding agrees earlier definition using syzygies example veronese example construct explicit element cohomology space kernel defines element turn proof theorem corollary convex lattice polytope one associate inner normal fan whose rays correspondence facets prime divisors general divisor vector space basis naturally corresponds set rays fan multiplication global sections corresponds coordinatewise addition lattice points divisor whose global sections correspond gives ample line bundle embedding projective space note setting nothing changes extending field next proof assume algebraically closed also use following proof taking line bundle birational morphism projective normal varieties preserves global sections also use fact adding divisor amounts taking interior corresponding polytope proof theorem using results next section rely koszul duality requires smoothness letp toric resolution singularities let canonical divisor let line bundle coming projective embedding also denote globally generated hence nef demazure vanishing koszul duality first equality follows taking preserves global sections claim dimension kernel following map resp mean vector space resp basis corresponds know note image contained results theorem follow theorem corollary theorem except result easy case note theorem equals theorem duality proof theorem follows whenever proof theorem denote dimension kuszul cohomology graded module opn let number lattice points standard simplex dimension proof theorem duality strictly speaking apply theorem proof obviously generalizes vanishing condition satisfied line bundle vanish dimension kernel applying theorem find whenever equivalently formula follows corollary combinatorial proof section require polytopes normal instead working spaces monomials etcetera replace monomials xinn corresponding lattice point let finite let abusively write resp vector space resp basis interested kernel following map vector space zero mean addition definition let finite subsets integer uniquely written order linear combination coefficients expressions form total order define support denoted supp convex hull set points occurring wedge part terms definition lattice reflexive transitive relation call lattice order one way obtain lattice take linear map set coefficients defining linearly independent defines lattice order write write proof following lemma use property points either lemma let finite subsets let ker let lattice supp unique maximum let supp set points defined analogously finally let exists ker terms containing part proof write terms containing part without redundant terms define prove ker clear supp let prove every expression know maximal applying term nothing cancel contradicting fact reason nothing cancel terms end unless unique maximum supp implies still one term ending nothing cancel domain prove terms containing part terms cancel terms without part must zero example veronese example becomes new notation explicit cochain becomes take coming linear map unique maximum supp lemma gives indeed last term wedge part supp theorem notation suppose injective proof induction case trivial domain zero let take element kernel let lattice order apply construction lemma obtain ker since applying induction hypothesis get contradiction note also follows froml green linear syzygy theorem theorem applied graded module graded ring give direct proof rely technique later want construct elements kernel end following construction let consider elements variables monomial degree variables associate element kernel let list points point occurs times list let list points define sgn sign independent choice lists lemma constructed integer coefficients kernel proof consider permutations previous definition claim terms corresponding equal wedge product thing changes order change sign caused changing order compensated change sgn number bijections property equal hence expression integer coefficients prove kernel obviously sums claim applying everything cancels let set ordered pairs sgn partition unordered pairs belong pair either equal following conditions met conditions imply transposition hence sign one easily sees pairs yield terms cancel example suppose take example monomial take lists get course last term zero note term form belongs lower right blue triangle upper left one example suppose consider monomial get case first two wedge factors term left blue square third wedge factor right blue square definition terms term occurs twice divide two twelve terms left one way get rid factor sum one element equivalence class two permutations equivalent construction works field proposition distinct monomials linearly independent support linear combination convex hull union supp occurring coefficient proof induction case one monomial namely constant monomial corresponding unique point statement obvious suppose let distinct prove supp xaj supp conv prove equality enough prove every linear map attains maximum sides equation enough show injective dense given let order induces let maximum order greatest point occurring variable monomial prove maximum sides equation proving attains maximum sides obviously nothing greater possibly occur supp xaj xaj xaj occur containing define monomial degree using associate element xbj xaj xbj plus terms everything part smaller induction xbj maximum supp xaj finally xbj terms without part induction linear combination xbj zero maximum supp far studied kernel map introduce following maps time look intersection kernels nothing intersect put ker introduce new machinery helps prove main result example let ker ker proposition injective map sgn ker ker proof note definition depend choices injective cancellation impossible let prove last assertion define analogously prove follows ker ker let compute sgn sgn maps every number plus one everything smaller mapped every formally put bijection mapping note sgn sgn using bijection one sees proves hence ker ker anyt sequence points one defines element ker sgn list points whenever use notation assume intended basis ker intended basis ker lemma consider right group action set sequences permutation given sequence let monomial sum right cosets stabilizer proof enough prove equality characteristic zero sgn sgn sgn last equality follows theqorder stabilizer result follows removing factor ker generated lemma span intersected particular basis ker basis ker proof right group action restricting one ker maps sgn clearly element ker fixed action action set sequences previous lemma compatible action sense choose element orbit action sequences consider element ker linear combination prove generated write linear combination applying action expression permutes since fixed action linearly independent lemma depend linear combination note used previous lemma last equality proves first assertion second assertion follows first injectivity proposition established connection context context focus latter definition ker define suppi convex hull set lattice points occurring factor term following lemma analogous lemma lemma let ker let lattice fix suppose suppi strictly greater point suppi let set points let defined analogously write place place removed place ker omit proof since analogous lemma example let ker ker take order coming linear map unique maximum applying lemma get lemma ker proof analogous theorem lemmap let usual maps combination coefficients let linear suppi convex hull suppi particular linearly independent one prove technique proposition lemma let lattice polytope dimension least two lattice points let usual maps ker suppi suppi note lemma fail conclusion lemma fails proof prove tpby induction suppose without loss generality let ker take integer coordinates let using unimodular transformation suppose case contains one lattice point take linear map attain maximum take attain maximum one point induces lattice apply lemma place obtain ker get contradiction induction hypothesis get contradiction lemma needed fact contains one lattice point ensure dimension least two apply induction hypothesis case contains one lattice point suppose lattice point note define follows claim resp attains maximum one lattice point indeed applying lemma resp place get resp case obtain leads contradiction lemma unless resp attain maximum unique point case apply lemma therefore resp attains maximum one lattice point case let points reaches maximum let points reaches maximum know maximality similarly case get contradiction fact case let bepthe projection deletes first two coordinates maps zero otherwise cancellation different therefore define linear automorphism else recall resp mean vector space resp basis define linear map basis elements linearly extend base field define else follows ker ker define else therefore ker suppp suppp define suppp convex hull points occur first factor term course prove whenever support contains point resp attains maximum perform reasoning case suppp obtain contradiction choose resp high enough resp attain maximum points suppp means least two points suppp resp attains maximum since suppp suppp true suppp done theorem convex bounded lattice polytope dimension greater one tpthen expressions basis ker hence monomials degree variables basis ker proof let set bounded convex lattice polytopes either dimension greater one two lattice points lemma prove tpthe first statement fact prove generate ker lemma prove induction suppose first two points show kernel generated expressions form consider map sets given addition lattice points every point reached two elements namely write direct sum linear span kernel direct sum kernels restricted span result easily follows induction step suppose let extreme point conv using unimodular transformations one squeeze way points first coordinate smaller also make sure smallest first coordinate zero one follows first one chooses linear form attains maximum one choose integer coefficients prime factors share one chooses unimodular transformation whose first component first coordinate greater point next one chooses unimodular transformation form large enough coordinates smaller points finally one uses translation map greatest first coordinate minus smallest claim enough prove statement case suppose true prove arbitrary proof ker linear combination lemma supports contained supports hence henceforth assume put lexicographical ordering meaning smallest suppose exists ker linear combination translate first factor take lexicographic maximum minimal lexicographic infinite descents find contradiction let maximum first coorn dinate let point first coordinate zero lemma follows first coordinate coordinates equal apply lemma obtain ker usual maps satisfies plus terms whose first factor induction using fact xpm plus terms whose first factor smaller see maximum xpm terms cancel maximum smaller contradicting minimal choice fact important ensures last assertion theorem follows lemma corollary let convex lattice polytopes dimension greater usual map dimension ker follows number degree monomials variables end case time formula works theorem let conv let bounded convex lattice polygon dimension kernel usual map proof let put lattice order let first coordinate define abusively write power whose exponent first coordinate expressions kernel prove basis kernel proves theorem exactly induction case easy domain kernel points basis suppose let ker show linear combination expressions like let maximum supp apply induction hypothesis done assume supp lemma plus terms containing part write note also otherwise terms removed would nothing cancel would terms point agrees first coordinate applying induction hypothesis get therefore written plus terms containing part apply induction hypothesis conclude linear combination expressions like finally show linear independence expressions induction case trivial let let linear combination expressions yields zero containing support written plus terms containing sign expression like instead set stead induction hypothesis linearly independent follows occur wedge part otherwise linear combination could yield zero one applies induction hypothesis obtain contradiction references marian aprodu jan nagel koszul cohomology algebraic geometry university lecture series american mathematical society winfried bruns aldo conca tim koszul homology syzygies veronese subalgebras mathematische annalen wouter castryck filip cools jeroen demeyer alexander lemmens computing graded betti tables toric surfaces preprint wouter castryck filip cools alexander lemmens canonical syzygies smooth curves toric surfaces preprint david cox john little hal schenck toric varieties graduate studies mathematics american mathematical society lawrence ein daniel erman robert lazarsfeld quick proof nonvanishing asymptotic syzygies algebraic geometry lawrence ein robert lazarsfeld asymptotic syzygies algebraic varieties inventiones mathematicae david eisenbud geometry syzygies second course commutative algebra algebraic geometry graduate texts mathematics new york francisco gallego bangere purnaprajna results rational surfaces fano varieties journal die reine und angewandte mathematik ornella greco ivan martino syzygies veronese modules communications algebra milena hering syzygies toric varieties thesis university michigan frank loose graded betti numbers plane algebraic curves manuscripta mathematica elena rubei result resolutions veronese embeddings annali dell ferrara sezione vii scienze matematiche giorgio ottaviani raffaela paoletti syzygies veronese embeddings compositio mathematica euisung park syzygies veronese embedding arbitrary projective varieties journal algebra hal schenck lattice polygons green theorem proceedings american mathematical society schreyer syzygies canonical curves special linear series mathematische annalen
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acting thoughts towards mobile robotic service assistant users limited communication skills nov burget fiederer kuhner aldinger schirrmeister boedecker nebel ball burgard autonomous service robots become affordable thus available also general public growing need user friendly interfaces control robotic system currently available control modalities typically expect users able express desire either touch speech gesture commands requirement fulfilled majority users paralyzed users may able use systems paper present novel framework allows users interact robotic service assistant fashion using thoughts interface bci system composed several interacting components neuronal signal recording decoding task planning motion manipulation planning well environment perception various experiments demonstrate applicability robustness real world scenarios considering tasks tasks involving interaction results demonstrate system capable adapting frequent changes environment reliably completing given tasks within reasonable amount time combined planning autonomous robotic systems interesting new perspectives open humanrobot interactions ntroduction patients heavily impaired communication capabilities severly paralyzed patients condition forces constantly rely help human caretakers robotic service assistants autonomy patients offer adequate interfaces possess sufficient level intelligence generally intelligent system requires adaptive task motion planning modules determine appropriate task plans motion trajectories robot implement task real world moreover requires perception component detect objects interest avoid accidental collisions obstacles typically used interfaces haptic buttons audio speech visual gesture interfaces command robotic system intuitive easy options healthy users difficult impossible use paralyzed individuals paper present novel framework schematically depicted fig allows interaction users minimal communication capabilities motion manipulation planning motion execution planner knowledge base user interaction object detection fig framework unifying decoding neuronal signals task planning motion manipulation planning scene perception centralized knowledge base core intuitive goal selection provided adaptive graphical user interface robotic service assistant record neuronal activity elicited human brain common origin types communication electroencephalography eeg system furthermore adopt convolutional neural network approach online decoding neuronal activity order allow users navigate graphical user interface gui provided task planner set feasible actions displayed gui depends turn current state world stored central knowledge base continuously updated information provided robot camera perception system task selected decomposed sequence atomic actions planner subsequently action resolved motion mobile manipulator using motion manipulation planning techniques following individual components shown fig described detail presenting quantitative evaluation overall system regarding performance elated ork authors contributed equally work authors department computer science faculty medicine university freiburg germany burgetf kuhnerd aldinger jboedeck nebel burgard research supported german research foundation dfg grant number exc grant stiftung multiple previous studies focused robotic systems assisting people disabilities example park implemented system autonomous feeding yogurt person chung focused autonomous drinking involved locating drink picking bringing person mouth using hybrid bci head movement control achic studied setup moving wheelchair attached robotic arm none systems used pure bci control contrast wang used motor imagery bci three classes achieve control robotic arm relevant developed robotic system receives bci command user autonomously assists user drinking cup however approach considers single object manipulator recently muelling presented approach assistive robotics albeit focused invasive bcis nonetheless approach could combined planning approach presented work applications robust decoding brain signals required inspired successes deep convolutional neural networks convnets computer vision speech recognition deep convnets recently applied frequently eeg decoding deep convnets already applied decoding tasks useful building interfaces lawhern used deep convnet decode oddball signals feedback negativity two tasks evaluated trained subjects evaluated others convnet yields competitive accuracies compared traditional decoding algorithms tabar halici used convnet combined convolutional stacked autoencoder decode motor imagery yielding better accuracies several decoding algorithms schirrmeister used shallow deep convnet decode motor imagery motor execution reaching slightly surpassing accuracies widely used eeg algorithm filter bank common spatial patterns bashivan used convnet trained inputs estimate mental workload addition work evaluating convnet decoding accuracies convnet visualization methods allow get sense features network using taken together advances make deep convnets viable alternative brainsignal decoding interfaces still knowledge online control deep convnets yet reported interface iii nline ecoding euronal ignals system hand developed control complex scenarios ones considered previous work particularly consider scenarios involving manipulation objects well interaction feasible goals determined gui controlled directional commands reliable classification brain signals navigation directions yet achieved directly bcis used deep convnet approach decoding multiple mental tasks eeg schirrmeister approach introduces hybrid network combining deep convnet shallower convnet chitecture deep part consists blocks using exponential linear units elu max pooling whereas shallow part uses single convolutionpooling block squaring mean pooling parts use final convolution elu produce output features features concatenated fed final classification layer trained convnet decode five mental tasks right hand finger feet toe movements object rotation word generation rest mental tasks evoke discernible brain patterns used surrogate signals control gui offline training done cropped training strategy using shifted time windows within trials input data experience important train bci decoder subjects environment close possible real application environment avoid pronounced performance drops therefore designed gradual training paradigm within planner gui displayed environment timing actions identical real control task training paradigm proceeds follows first train subject offline using simulated feedback subjects aware control gui mental tasks cued using grayscale images presented center display times fixation circle displayed center gui subject instructed fixate minimize eye movements random time interval fixation circle switched disk indicates end mental task time gui action select back nothing corresponding cued mental task performed update gui keep training realistic include error rate average every fifth action erroneous instruct subjects count error occurrences assert vigilancy offline data used train individual deep convnets subjects online training performing decoded mental tasks gui finally stop cueing mental tasks evaluate performance bci control let subjects create instructed plans gui tasks executed simulated robot real mobile manipulator available provide control mobile manipulator enhance feeling agency subjects confirm execution every planned action interrupt chain actions moment execution bci decoding accuracies instructed tasks assessed manually rating decoding based instructed task steps statistical significance decoding accuracies tested using conventional permutation test random permutations labels fraction label permutations would led better equal accuracies accuracy original labels igh evel oal ormulation lanning use domain independent planning derive required steps reaching desired goal complex task user formulate goal without knowledge internal representation objects planning system exact capabilities robot achieved intuitive graphical user interface object parameters goal specified incrementally refining objects referring type cup attributes content domain independent planning identifies sequence actions transforms current world state state satisfying goal condition planning task consists planning domain describing static components object type hierarchy available actions problem instance describing objects present world current state well goal description current state objects extracted knowledge base goal chosen gui restricted vocabulary shared user planning system objects sets objects identified creating referring expressions composed shared references built vocabluary briefly describe relevant aspects previous work area general referring expression logical formula single free variable refers object valid reference cup contains water refers cups containing water restrict references simple conjunctions facts preferable computational reasons also allows incrementally refine references adding constraints example adding contains water cup restricts set cups set cups containing water distinguish three types fundamental object references individual references typename references relational references individual references identified name omnirob robot typename references identified name type refer cups scenario directly refer unspecific cup relational references encountered objects referred via predicates occur argument relations scenario object attributes example content cup used clarify cup meant object references used create references goals start defining goals action achieves found natural user put cup shelf initial selection goal type drop necessary determine objects parameters goal predicate action parameters refined constraining previous choice argument either determined uniquely impossible constrain argument user declares remaining option acceptable exclude unreachable goals allow goals achieved sequence preceding actions drinking water could require fetch cup bring patient fetch bottle pour water cup order executed goal determined selection process gui passed custom domain independent planner robot otion eneration generating paths mobile base apply planning framework rrt given pair terminal configurations performs bidirectional search using uniform sampling configuration space initial solution path found path subsequently refined remaining planning time adopting informed sampling strategy yields higher rate convergence towards optimal solution execution paths implemented via joint trajectory tracking algorithm using robot localization feedback realize pick place pour drink motions efficiently adopt probabilistic roadmap planner approach planner uses graph randomly sampled task poses endeffector poses connected edges find plan two poses planner connects poses roadmap graph uses algorithm find optimal path start goal pose execution plan maps task space velocity commands joint velocity commands employing task space motion controller sample random poses around object determine grasp motions dropping objects extract horizontal planes camera point cloud sample poses planes find suitable drop location mplementation etails implementation framework real world requires several components neuronal signal decoding scene perception knowledge base operations well symbolic motion planning run parallel therefore distributed computation across network computers communicating among via ros decoding neuronal signals four components eeg measurements performed using waveguard eeg caps electrodes neurone amplifier mode additionally vertical horizontal eogs emgs four extremities ecg recorded recording used matlab transferred data gpu server deep convnet classified data classes planner gui consists backend gui uses fast downward planner iteratively build goal references find symbolic plans selected goal planning time crucial performance system used fast downward basic configuration experiments central knowledge base implemented ros node able store objects arbitrary attribute information changes knowledge base automatically trigger updates unexpected ones interrupt current motion trajectory execution finally used simtrack object pose detection tracking vii xperiments evaluate framework consider environment schematically depicted fig containing two shelves table potential locations manipulation actions table aggregated results bci control runs value runs accuracy shelf shelf robot user gui table eeg camera object location fig experimental environment two shelves table considered robot performing manipulation actions five rgbd sensors observe environment human operator selects goal using eeg control planner gui user sits wheelchair front screen displaying graphical interface planner robot used experiments omnirob mobile manipulator platform kuka robotics composed degrees freedom dof dof mobile base dof manipulator additionally dexterous hand schunk attached manipulator flange used perform grasping manipulation actions tasks considered experiments required robotic system autonomously perform following actions drive one location another pick object drop object shelf table pour liquid bottle cup supply user drink moreover use perception system composed five rgbd cameras three statically mounted shelves table order observe scene report captured information knowledge base two cameras carried robot first one located mobile base used perform collision checks manipulation planning second camera mounted robot used tasks involving physical interaction demonstration framework found accompanying video http online decoding neuronal signals evaluated bci control setup four healthy subjects three females aged time writing validation still progress validation mobile manipulator performed total runs recorded real robot subjects executed various instructed plans runs used simulated feedback gui order generate significant amount data evaluation performance bci decoding runs assessed using video time steps path optimality recordings interactions gui rated gui actions correct correspond instructed path incorrect otherwise actions necessary remediate previous error interpreted correct correction intentionally clear finally rated rest actions correct simulated robot executions incorrect next robotic action initialized ignored plan creation evaluation five metrics extracted video recordings accuracy control time took subjects execute plan iii number steps used execute plan path optimality ratio steps used minimally possible number steps average time per step summarized results table total correct bci control achieved required per step selecting plan using gui took average required user perform average steps gui highlevel planner path formed steps average away optimal path decoding accuracy every subject significantly chance eeg data used train hybrid convnets decoding results transfer visualized fig fig show ratio snr classes labeled datasets define snr given frequency time electrode snrf iqr median median iqr corresponds set values position task number repetitions median iqr median interquartile range iqr respectively upper part describes variance class medians large variance means distinguishable class clusters higher snr denominator describes variance values class lower variance values results higher snr low snr emg channels shows subjects move tasks decoding accuracies achieved test data initial training convnets visualized fig support neural origin bci control signals fig shows physiologically plausible correlation results see methods table aggregated results runs actions snr executions scheduled success executions grasp drop approach total table iii aggregated results runs actions grasp drop approach pour drink hand feet predictions runtime mean std hand rotation feet words rotation rest words sensitivity targets rest correlation fig eeg data decoding results snr first data used train hybrid convnet highest snr observed alpha lower beta bands frequency bands robust markers task related mental tasks note channels top row withheld convnets time displayed negative control channels displayed space constraints confusion matrix decoding accuracies transfer accuracies well theoretical chance level topographically plausible correlation maps alpha frequency band details visualization technique refer reader total executions scheduled success executions runtime mean std might differ two reasons number executed calls lower scheduled ones indicates previous action step failed succeed plan recovery possible hand higher number executed calls indicates user able achieve plan recovery commanding repetition failed action moreover recorded largest standard deviation approach action attributed diverse complexity planning problem mobile base distance travel selected grasp drop location total system achieved success rate entire task planning execution required average errors mainly caused object detection issues system able detect object detection precise enough able successfully grasp drop object drinking task fetch carry task first experiment considering use real robot evaluates complete system tasks goal transfer object one location another shelf table using robot fulfill tasks robot typically needs execute four subtasks approach object location grasp object approach location drop object user instructed select predefined goal using planner moreover selected random initial placement objects run order cover different environment states experiment repeated ten times user table shows averaged results experiment second column indicates overall number desired action calls scheduled planner well number calls actually performed third fifth columns represent success rate mean standard deviation runtime actions respectively note number scheduled actually executed actions last experiment evaluates direct interaction user robot therefore implemented autonomous robotic drinking assistant approach enables robot fill cup liquid move robot user finally provide drink user execution corresponding drinking motion front user mouth addition techniques described successful pouring drinking using robot requires detection liquid level cup reliable detection localization user mouth detect liquid level pouring follow approach introduced given camera viewing angle liquid index refraction liquid height determined depth measurement using relationship based snell law see details using knowledge first detect cup extract depth values liquid finally estimate real liquid height type liquid hence index refraction assumed given beforehand viewing angle determined depth data kalman filter used track liquid level compensate noise detected liquid level exceeded user defined value stop signal sent terminate pouring motion detection localizing user mouth adopt two step approach first step segment image based output face detection algorithm order extract image region containing user mouth eyes afterwards detect position mouth user considering obtained image patch regarding mouth orientation additionally consider position eyes order obtain robust estimation face orientation hence compensating slightly changing angles head face mouth eye detectors implemented opencv applying algorithm uses haar cascades table iii shows averaged results experiment scheduled actions repeated order complete task successfully one run plan recovery possible leading abortion task thus system achieved total success rate drinking task planning execution required average evaluation liquid level detection approach specified desired fill level executed runs pour action resulting mean error standard deviation instances bottle obstructed camera view resulting poor liquid level detection higher error viii onclusions paper presented mobile robotic service assistant capable successfully performing complex tasks including close range interaction user continuously changing environment increase independence severely paralyzed patients use planner intermediate layer user autonomous mobile robotic service assistant overcome curse dimensionality typically encountered bci control schemes thus opening new perspectives interaction scenarios eferences park kim erickson kemp towards assistive feeding mobile manipulator chung wang cooper autonomous function robotic manipulators perform daily activities rehabilitation robotics icorr ieee international conference ieee achic montero penaloza cuellar hybrid bci system operate electric wheelchair robotic arm navigation manipulation tasks advanced robotics social impacts arso ieee workshop ieee wang xia yang xiao velez yang motor imagery robot arm system natural computation icnc seventh international conference vol ieee killmann frank fiederer ball burgard autonomous robotic assistant drinking robotics automation icra ieee international conference ieee muelling venkatraman valois downey weiss javdani hebert schwartz collinger bagnell autonomy infused teleoperation application brain computer interface controlled manipulation autonomous robots krizhevsky sutskever hinton imagenet classification deep convolutional neural networks advances neural information processing systems zhang ren sun deep residual learning image recognition sainath kingsbury saon soltau mohamed dahl ramabhadran deep convolutional neural networks speech tasks neural networks vol sercu puhrsch kingsbury lecun deep multilingual convolutional neural networks lvcsr ieee international conference acoustics speech signal processing icassp lawhern solon waytowich gordon hung lance eegnet compact convolutional network interfaces corr vol tabar halici novel deep learning approach classification eeg motor imagery signals journal neural engineering vol schirrmeister springenberg fiederer glasstetter eggensperger tangermann hutter burgard ball deep learning convolutional neural networks brain mapping decoding information human eeg ang chin zhang guan filter bank common spatial pattern fbcsp interface ieee international joint conference neural networks ijcnn bashivan rish yeasin codella learning representations eeg deep neural networks arxiv stober learning discriminative features electroencephalography recordings encoding similarity constraints bernstein conference clevert unterthiner hochreiter fast accurate deep network learning exponential linear units elus arxiv vol dale reiter computational interpretations gricean maxims generation referring expressions cognitive science vol assisting goal formulation domain independent planning advances artificial intelligence springer burget bennewitz burgard rrt efficient path planning framework mobile manipulation international conference intelligent robots systems iros daejeon korea kavraki latombe probabilistic roadmaps robot path planning john wiley helmert fast downward planning system journal artificial intelligence research jair pauwels kragic simtrack framework scalable object pose detection tracking international conference intelligent robots systems iros ieee schubert burgard probabilistic approach liquid level detection cups using camera international conference intelligent robots systems iros daejeon korea hara honda tsubouchi ohya detection liquids cups based refraction light depth camera using triangulation iros viola jones rapid object detection using boosted cascade simple features computer vision pattern recognition cvpr proceedings ieee computer society conference vol ieee lienhart maydt extended set features rapid object detection image processing proceedings international conference vol ieee
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jul machine learning deepest learning statistical data assimilation problems henry abarbanel department physics marine physical laboratory scripps institution oceanography center engineered natural intelligence habarbanel paul rozdeba sasha shirman department physics university california san diego gilman drive mail code jolla usa draft july abstract formulate strong equivalence machine learning artificial intelligence methods formulation statistical data assimilation used widely physical biological sciences correspondence layer number artificial network setting analog time data assimilation setting within discussion equivalence show adding layers making network deeper analogous adding temporal resolution data assimilation framework one find candidate global minimum cost functions machine learning context using method data assimilation discussed calculations simple models side equivalence reported also discussed framework time layer label taken continuous providing differential equation equation shows problem solved two point boundary value problem familiar discussion variational methods use continuous layers denoted deepest learning problems respect symplectic symmetry continuous phase space lagrangian versions hamiltonian versions problems presented implementation discrete respected symplectic structure addressed hamiltonian version provides direct rationale back propagation solution method canonical momentum introduction use enhanced computational capability two seemingly unrelated inverse problems flourished past decade one machine learning realm artificial intelligence developments often name deep learning data assimilation physical life sciences describes transfer information observations models processes producing observations paper directed towards demonstrating two areas investigation fundamental level statistical physics problem methods utilized one may prove valuable main goal paper point many developments data assimilation utilized arena machine learning also suggest innovative methods machine learning may valuable data assimilation two areas focus attended variational annealing method action cost function machine learning statistical data assimilation permits location apparent global minimum cost function notion analyzing problem continuous time layer call deepest learning wherein clear one addressing two point boundary value problem underlying symplectic structure methods abound solving two point boundary value problems assuring symplectic structures respected time layer discretized may quite fruitful machine learning paper primarily discusses multilayer perceptrons feedforward networks though also makes clear discussion carries recurrent networks well background machine learning standard feedforward neural nets begin brief characterization simple architectures feedforward neural networks network describe composed input layer output layer hidden layers within layer active units called neurons usually chosen layer neurons degrees freedom layer purposes neurons layer structure generalized different numbers different types neurons layer cost notation explosion data available layer layer pairs input output sets vectors labels pairs index data activity units hidden layer determined activity previous layer index combines neuron number neuron one label connection described nonlinear function via summation weights determines activities layer combined allowing act yielding activities layer numerous choices manner weight functions act well numerous choices nonlinear functions direct reader references discussion virtues choices input output layers network activities compared observations network performance assessed using error metric often least squares criterion cost function permits inhomogeneous weighting comparison network activities data minimization cost function weights subject network model used determine weights variables layers parameters appearing architecture network moving along data assimilation note one wishes find global minimum cost function nonlinear function neuron activities weights parameters functions layer problem suggests one find global minimum machine learning problem set unless special circumstance see circumstance data assimilation problem equivalent machine learning tasks machine learning problem described assumes error model minimization cost function subject strong equality constraints model results output layer complicated function parameters model activities layers likely connected many local minima associated nature search problem introduce variational annealing method next section regularizes moving equality constraint cost function via penalty function introduces hyperparameter allowing systematically vary complexity search process standard statistical data assimilation describe formulation statistical data assimilation problem data assimilation observations made sparse set dynamical variables associated model processes producing observations allow estimation parameters model unobserved state variables model goal estimate unknown parameters model dynamical state variables model may observed estimate unmeasured state variables well certain time window information transferred model estimate full model including initial condition state variables predictions made completed model compared new observations validation prediction essentially question generalization addressed machine learning data assimilation one window time observations made times lie observation time measurements made knowledge measurement instruments observations related state variables model via measurement functions using knowledge processes producing observations develop dynamical model state variables satisfy set dynamical differential equations dxa time dependence vector field may come external forcing functions driving dynamics set differential equations necessarily represented discrete time solving numerically resulting map discrete time vector field related via integration procedure one chooses starting initial condition use discrete time model move forward first observation time eventually reaching end observation window altogether making model integration steps intervals make time steps measurements noisy models errors statistical problem goal construct conditional probability distribution model states conditioned measurements assuming transition model state time depends model state time dynamics markov using identities conditional probabilities one may write action log suppressing dependence observations log log conditional mutual information given log model error free delta function representation may evaluate conditional expected values functions path model observation window exp exp log log log terms depending observations canceled numerator denominator expected value observations times independent measurement function identity noise measurements gaussian diagonal inverse covariance matrix first term action measurement error term takes form measurement made error model taken additive gaussian diagonal inverse covariance matrix second term model error term becomes term constants normalizations gaussians cancel expected value keep constants take limit would restore delta function dynamics finally one accepts ignorance distribution initial conditions selects uniform dynamical range model variables evaluated exp exp desired connection machine learning formulation model error statistical data assimilation formulation identify layer labels time alone could passing interest however much connection may utility discuss items data assimilation language translation easy point statistical data assimilation call standard model critical suggestion relative standard practice machine learning allowing finite acknowledging model may add additional tool exploration machine learning environments typically account model error introduced serves regulating device complexity surface path space estimation states parameters occurs data assimilation developments use machine learning finding global minimum key establishing estimates unobserved state variables unknown parameters model perform approximately course integral one using monte carlo methods method laplace laplace method one seeks minima action integral gaussian would functions nonlinear must perform numerical evaluation laplace method approximates integral contributions lowest minima action one find minima associated paths larger action give exponentially smaller contributions expected values paths smaller action allows one circumvent search global minimum parameters hyperparameters aspects model data yield set action levels connected minima action one path yields action level much smaller path numerical accuracy one may use smallest minimum path comprised parameters hidden unobserved state variables neglecting larger minima action developed variational annealing approach finding path smallest value action proof global minimum found numerical results indicate may case method produces set minima action giving numerical clue roughness surface path space data assimilation surface depends among items number measurements observation time number model time steps measurement times translates directly analogous machine learning problem time layer number model time steps measurement times increases number hidden layers increases model architecture deepens proceeds kind numerical continuation requirement varying parameters minimizes procedure begins taking namely complete opposite value found usual machine learning deterministic error free layer layer maps outset limit action quadratic function model variables times measurements made minimization simple data presented input output layers minimum degenerate know values state variables first step choose solution optimization problem select states drawn uniform distribution ranges known dynamical range state variables one learn well enough solving underlying model forward various initial conditions make draw times paths candidates procedure select small value call previous paths initial choices minimization algorithm find paths minimization problem gives values action associated new paths next increase value found values range good choices new value perform minimization action starting initial paths previous step arrive new paths evaluating action paths gives ordered set actions longer degenerate many paths may give numerical value action however typically degeneracy lies within noise level data procedure continued large enough indicated least one action levels becoming substantially independent check calculation observe action independent expected value measurement error term measurement errors taken gaussian term action distributed expected value readily evaluated action levels expected value large procedure consistent increases required effectively starts problem global minimum apparent systematically tracks many paths increases tracking global minimum one must check selected value large lest one leave global minimum land another minimum checking result using smaller worthwhile important note simply starting large value places one undesirable situation action multiple local minima optimization procedure quite likely fall dynamical problems examined one typically finds number measurements increased fewer fewer minima action remain large enough one minimum attribute additional information augmented set measurements manifest discussion additional information effectively controls unstable directions phase space smallest minimum necessarily convex action goal provide accurate estimations conditional expected value functions path model space distributed exp actually require convexity function path space point view accurately estimating expected values sufficient lowest action level much smaller second lowest action level action value lowest level xlowest much smaller action value next minimum xsecond lowest factor exp xlowest xsecond lowest lowest path xlowest dominates integral done provides sensible choice path evaluate integral examples feedforward neural networks data assimilation section examine one example perceptrons one example statistical data assimilation latter utilizes differential equation model introduced lorenz permits one easily increase number dimensions phase space easily select number observations within designated measurement window easily choose number model evaluations measurement times latter analogous increasing number layers perceptron case perform twin experiment use model generate solutions initial conditions solutions gaussian noise added become noisy data using noisy data use estimate unobserved state variables hidden layer variables data assimilation model begin examining dynamical equations introduced dxa fixed parameter take solutions dynamical equations chaotic equations states meant describe weather stations periodic spatial lattice model widely used atmospheric science testbed exploration innovative data assimilation ideas example selects displays action level plot observations measurement time within window perform twin experiment wherein generate time series using standard adaptive fourth order algorithm time step initial condition drawn uniform distribution range variables namely solutions add gaussian noise mean zero variance time series noisy versions model time series constitute data time series presented model times measurement window measurements made time step measurement error matrix taken diagonal elements measurement time zero times magnitude taken model error matrix also taken diagonal elements along diagonal performing procedure take chosen minimizations nonlinear objective functions example using model performed using public domain software ipopt front end script written python fig display action level plots observations measurement time see topleftpanel numerous local minima action values remain large none minima far separated paths smallest minimum evaluation expected value integrals would require contributions many maxima conditional probability distribution toprightpanel begin see isolated action level whose contribution expected value integral overwhelmingly larger contribution path giving rise next largest action level value consistent observation around instabilities state space appear controlled data assimilation process bottompanels see dominance lowest action level even clearer horizontal olive colored line expected value measurement error term action sign consistency data assimilation calculations fig explore another aspect action level plots still use hold fixed number observations within window reduced move model forward observations times provide analogy many layers present equivalent machine learning example example differs many entry points measurement window machine learning example one display leftpanel action level plots selected number model evaluation steps one see intermediate steps many persisting minima action intermediate steps single minimum found large comes action level intermediate steps consistent expected value measurement error term calculation performed machine learning context provides information many hidden layers required achieve desired accuracy rightpanel display accuracy estimation single parameter model set producing data value clearly selected intermediate model evaluations clearly selected intermediate steps zero intermediate steps consistent percent error see collection calculations noted earlier ability identify dominant minimum action depends number measurements presented statistical data assimilation procedure embodying transfer information data model data assimilation associated number positive conditional lyapunov exponents model machine learning instantiation may play role number data presented output layer sufficient determine parameters hidden states layer also see analogue deepening network produces higher accuracy estimates conditional expected values perceptron feedforward network constructed feedforward network layers one input layer one output layer network hidden layers analyzed neurons layer activity neuron layer related previous layer tanh also investigated relu function log report results activations input layer drawn gaussian weights selected uniform distribution gaussian measurement noise added data generated model zero mean variance figure action level plots model variational annealing procedure performed different initial conditions minimization action value top left panel measurements measurement time many minima none much smaller others dominates expected value integral top right panel measurements measurement time action path space numerous local minima lowest minimum action value much smaller action values minima dominates expected value integral bottom left panel measurements measurement time number minima found two lowest minimum dominates expected value integral bottom right panel measurements measurement time one minimum action solid green line expected value measurement error term distributed action becomes independent expected value equal value figure parameter estimation action level results model parameter value used twin experiments model observations made every measurements made observation time left panel action level estimates action levels observations made measurement time choice model evaluation steps measurement times horizontal olive green line indicates expected action level large right panel parameter estimates observations model evaluated times leading parameter estimates quite accurate model time steps observations distributed model step observations one associate idea increasing number model steps observations equivalent deepening hidden unobserved layers horizontal olive green line indicates parameter value used generating data data twin experiment know weights inputs generate values layer recorded gaussian noise mean zero variance added data layers starting known values data input layer output layer pairs presented model inputs layer data outputs layer investigated minimize action weights states layers model neurons layer data input output layers use variational annealing procedure described identify action levels various paths network initial value taken incremented via numerical optimizations machine learning example used least two ways present information model setting increase number training pairs available network number chosen large user wishes increase number components pair vectors number number neurons resolution model ability capture variations activity layer layer improved increasing number layers fig explore action levels function increase number layers model hold fixed number neurons number training pairs number inputs outputs case many local minima lowest action minimum significantly split action minima qualify dominate expected value integral increased lowest action minimum comes closer second lowest minimum seen result much larger number weights estimated latter value holding fixed values information available make estimations fig hold fixed fixed look values dimension pairs fig hold fixed number layers number neurons layer dimension vectors show effect increasing number pairs emergence lowest action minimum increases displayed serve candidate approximating fig display error prediction mode layers set estimation weights error constructed selecting new input output pairs using input elements components use model estimated weights evaluate compare pairs square error averaged presented components pairs lmp displayed see increasing information presented via increasing leads decreased average prediction errors choosing path corresponding lowest action level top layers choosing path associated second lowest action level bottom panel differences quality prediction generalization examples among cases analyzed large noted recurrent networks network architecture one allows interactions among neurons one layer another layer well interactions among neurons within figure holding number neurons layer fixed number pairs fixed number inputs outputs fixed vary number layers deepening network examine action level plots arising variational annealing procedure upper left panel top right panel bottom left panel bottom right panel figure holding number neurons fixed number layers fixed number training pairs fixed display action levels vary number inputs number outputs top left panel top right panel bottom panel figure holding number neurons fixed number layers fixed number inputs number outputs display action levels vary number training pairs top left panel top right panel bottom panel figure prediction errors learning network averaged new pairs noisy input produces output using model estimated using training sets compared output produced original model used produce data twin experiment case number neurons top left panel using model associated lowest action level top right panel using model associated lowest action level bottom panel using model associated second lowest action level lmp pmp single layer activity neuron layer given wji feedforward layer goes next layer network add interactions layer fashion give dynamics activity introduce sequential label activity neuron layer mapping layer layer within layer summarized wji wji another version allows nonlinear function different connections connections wji wji different nonlinear functions translate expressions structure recognizing model variable function recurrent network state variables become seems natural dimensions connectivity going solely feedforward plus additional independent variables would aspects neuron state variables representation adding connections among neurons within layer another independent variable called point neurons depending layer alone become fields machine networks restrictions number independent variables may lead investigation neural fields collection independent variables indicating layers involved progression field input output layer however many independent variables however many neurons utilize architecture model network overall goal identifying conditional probability distribution estimating moments expected values interest still comes one form another approximation integrals making time continuous continuous layers deepest learning much learn data assimilation machine learning problem number layers equivalently number time points within epoch becomes large limit action number layers epoch becomes continuous variable data assimilation notation formulation quantity near times taken proportional within machine learning context call deepest learning number layers goes infinity useful manner minimization action paths space satisfy equation along boundary conditions canonical momentum standard model equations take form dxb dfab equations necessary condition along accompanying boundary conditions show errors represented right hand side equation drive model variables layers produce data available integral coordinates restricted natural boundary conditions shows quite clearly minimization problem requires solution two point boundary value problem space one way address two point boundary value problems start one end value proceed value integrate ways requiring match furthermore residual measurement error term right hand side nudges solution desired output one specify boundary conditions equation given require canonical momentum examining hamiltonian dynamics problem suggest integrating equation forward canonical momentum equation backward back propagation hamiltonian dynamics realization one moves lagrangian realization variational problem hamiltonian version trading phase space canonical coordinates hamiltonian standard model reads coordinates equations motion given hamilton equations dpa dxa returning discrete time layers see variational principle carried space boundary conditions quite easy impose variables varied going forward backward neither required suggested formulation worth noting either space space continuous time layer formulation symplectic symmetry automatically maintained discrete time layer problem reinstated however many choices integration procedure becomes discrete symplectic symmetry maintained known detailed analysis variational problem lagrangian hamiltonian formulations appears direct lagrangian version state variables varied symplectic structure maintained boundary conditions canonical momentum respected practice means direct variational methods suggested machine learning problems taking account model error may skirt issues associated back propagation issue may seen bit directly comparing one moves space organized motion space guided equivalent motions model connected legendre transformation hamiltonian form limit one usually works moving regions may saddle points may slow progression canonical momentum may occur maximum minimum saddle point observation analysis seems show saddle points downward curving directions present large numbers almost similar values objective function hence much matter saddle points algorithm gets stuck may apply lagrangian formulation manner enters motion quite different may well avoid confounding property pointed backprop use lagrangian variational principle solves problem may unexpected virtue summary discussion paper directed drawing direct analogy formulation much utilized class machine learning problems set equivalent problems data assimilation encountered many physical biological geoscience problems well many engineering analyses data driven model development testing goal fundamental equivalence two inquiries core paper analogy allows identify methods developed data assimilation potentially quite useful machine learning contexts particular possibility using variational annealing produce global minimum action cost function standard model data assimilation observation error model error appears potentially value idea making time continuous purposes exploring properties data assimilation suggests similar tactic machine learning machine learning step making layers continuous called deepest learning deep learning appears result increasing number layers continuous layer time formulation see clearly problem solved two point boundary value problem may lead construction solution tractable models may helpfully illuminate deep learning networks operate successfully expand possibilities utilizing employing additional methods numerical calculations interpretation formulation statistical data assimilation problem general level expressed see measurement error term information data passed model explicitly information conditional mutual information passed observations model suggests idea deep learning works major increases computing power well large data sets however attribute data sets much large possess information precise manner utilized learn models conjunction information transfer state parameter estimation embodied work rissanen identifies cost estimating parameter state variable time arguments presented suggest evaluating much information data set available inform model greater utility size data set one point made explicit main text worth noting formulated data assimilation machine learning problems accurately performing high dimensional integrals laplace approximation method namely usual variational principle permits investigation corrections terms expansion action path leading global minimum shown corrections first approximation small becomes large analyzing standard model need case choices noise distributions measurement error model error terms action another item interest argument noted dimension model increases one may find fewer fewer local minima confounding search path space global minimum situation many unstable saddle points path space arise case chaotic system model evidence however large dimension model paths one may search multiple local minima number measurements observation time large enough information transferred model sufficient role number model evaluations observations suggested arguments also play significant part establishing whether action surface many local minima view deep network moving hidden layers many may also illuminated arguments one idea increasing number hidden layers one increasing resolution analog time data assimilation one data assimilation see probing variation underlying model evolves initial condition layer missing higher frequency variations time employing coarse grid discrete time counterpart role feedforward networks discussed recognized neuron models widely utilized machine learning applications little common properties biological neurons construction implementation large networks successful function within machine learning may prove useful guide construction implementation functional natural neural networks finally important comment analogy drawn utilized may improve testing validation models supported observational data assist selection models formulation still task addressed user acknowledgments express appreciation colleagues dan breen jeff elman important discussions also ceni team made finding analogs paper possible many thanks tim gentner gert cauwenberghs especially gabe silva improved manuscript jon shlens provided detailed view active world machine learning partial support muri program sponsored office naval research acknowledged support shirman arcs foundation references perspectives research artificial intelligence artificial general intelligence relevant dod https henry abarbanel predicting future completing models observed complex systems springer allgower georg numerical continuation methods introduction bennett inverse methods physical oceanography cambridge university press byrd nocedal limited memory algorithm bound constrained optimization siam journal scientific statistical computing anna choromanska mikael henaff michael mathieu rard ben arous yann lecun loss surfaces multilayer networks international conference artificial intelligence statistics aistats yann dauphin razvan pascanu caglar gulcehre kyunghyun cho surya ganguli yoshua bengio identifying attacking saddle point problem optimization proceedings international conference neural information processing systems nips pages cambridge usa mit press elman finding structure time cognitive science geir evensen springer data assimilation ensemble kalman filter robert fano transmission information statistical theory communication mit press gelfand fomin calculus variations dover publications ian goodfellow yoshua bengio aaron courville deep learning mit press cambridge london http ernst hairer gerhard wanner christian lubich geometric numerical integration algorithms ordinary differential equations springer series computational mathematics volume edition jordan attractor dynamics parallelism connectionist sequential machine proceedings eighth conference cognitive science society pages cognitive science society kadakia rey abarbanel symplectic structure statistical variational data assimilation quarterly journal royal meteorological society mark kostuk synchronization statistical methods data assimilation hvc neuron models phd dissertation physics university california san diego mark kot first course calculus variations american mathematical society providence rhode rubin landau manuel jose paez cristian bordeianu survey computational physics introductory computational science princeton university press pierre simon laplace memoir probability causes events physique tome pages pierre simon laplace memoir probability causes events statistical science translation english stigler yann lecunn yoshua bengio geoffrey hinton deep learning nature daniel liberzon calculus variations optimal control theory princeton university press edward lorenz predictability problem partly solved tim palmer renate hagedorn editors predictability weather climate cambridge marsden west discrete mechanics variational integrators acta numerica pages murty kabadi problems quadratic nonlinear programming mathematical programming alexander parlos kil chong amir atiya application recurrent multilayer perceptron modeling complex process dynamics ieee transactions neural networks press teukolsky vetterling flannery numerical recipes art scientific computing third edition cambridge university press jorma rissanen stochastic complexity statistical inquiry theory world scientific publishing river edge usa jorma rissanen information complexity statistical modeling springer wachter biegler implementation interiorpoint filter algorithm nonlinear programming mathematical programming wendlandt marsden mechanical integrators derived discrete variational principle physica nonlinear phenomena kadakia rozdeba abarbanel quinn improved variational methods statistical data assimilation nonlinear processes geophysics daniel rey nirag kadakia michael eldridge uri morone paul rozdeba henry abarbanel john quinn systematic variational method statistical nonlinear state parameter estimation physical review zhu byrd nocedal algorithm fortran routines large scale bound constrained optimization acm transactions mathematical software
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strongly connected components finding strongly connected components graphauthor munteanu abstraction directed graph strongly connected pair vertices exists path path computer science partition graph strongly connected components represented partition vertices graph two vertices partition exists path path two vertices different partition property met algorithm presented meant find partition given graph strongly connected components numberofnodes numberofedges log numberofnodes log function stands iterated logarithm algorithm better understanding algorithm recommended revision data structure section dfs oriented graph traversing section consider strongly connected component set doubt two strongly connected components represent two disjoint sets exists vertex belongs first strongly connected component well second strongly connected component means vertex first strongly connected component vertex second strongly connected component find path path holds belongs strongly connected component results two strongly connected components selected joined bigger one using induction prove two strongly connected components linked one vertices means jointed therefore strongly connected components graph pairwise disjointed initially consider every vertex strongly connected component definition vertex strongly connected component using dfs traversal link vertices strongly connected component algorithm recursive type following lines present steps must followed vertex added dfs stack marked visited either dfs source either adjacent vertex top vertex stack iterate adjacent vertices order every neighbor case already visited algorithm verifies vertex stack strongly connected component case positive joins strongly connected components case neighbor visited yet must introduced stack algorithm executed vertex starting first step execution algorithm popped stack visited checking mentioned nodes stack needs done case positive strongly connected components must joined vertex popped stack crucial point algorithm may know vertex stack connected components neighbor order solve problem must hold minimum level dfs stack strongly connected component initialized infinity level strongly connected component infinity vertex stack belongs reset infinity level strongly connected component vertex lowest level belongs component popped dfs stack two strongly connected components joined dominant one hold minimum level level level one proof correctness last paragraphs focused algorithm important observation allows use data structure following paragraphs focus proving correctness algorithm take account situations could appear adding new vertex stack let assume new added vertex vertex first pushed stack algorithm begins vertex father vertex belong strongly connected component vertex father vertex belong different strongly connected components case first pushed vertex father attempt change strongly connected component joining attempt done pairs father vertex case vertex father node found strongly connected component means exists path path direct edge consider path case valid paths consider first taken account dfs traversal due properties dfs traversal guarantees paths current node visited taken consideration could jump conclusion vertex vertexi visited vertices vertexi vertex vertexk visited dfs stack least one visited means chosen path first one selected dfs traversal would stack visited would mean complete dfs traversal executed previously visited added moreover vertex belongs path belong chosen path visited case previously visited either complete execution dfs traversal executed leads contradiction discussed chosen path first one first path one goes leads contradiction hypothesis therefore exists vertex vertexj property vertices vertexj vertex vertexk visited stack vertices vertex visited property dfs traversal guaranteed must lead traversal vertex point arrived vertex find vertexj visited stack strongly connected components vertex vertexj joined back recursion vertex confirmed fact vertex stack link vertexj inductively link would made vertex vertex vertex vertex finally stack vertexj conclusion two vertices connected edge component algorithm join case vertex father vertex different strongly connected components means exists path path exists direct edge exists path means vertex property exists path vertex property exists path exists path therefore dfs traversal would find path son father vertex set included connected vertex set included conclusion behavior algorithm desired possible cases correctness demonstrated complexity complexity given dfs traversal possible connection two vertices connected edge numberofedges log numberofnodes numberofnodes numberofedges log numberofnodes log stands iterated logarithm implementation simple clean implementation algorithm done five methods see code method generate set specific element vertex int getcomponent int currentelement int currentking currentelement currentscc currentking currentking currentking currentscc currentking currentelement currentking int copyofelement currentelement currentelement currentscc copyofelement currentscc copyofelement currentking return currentking method joins sets two given elements vertices void unitecomponents int firstelement int secondelement int firstcomponent getcomponent firstelement int secondcomponent getcomponent secondelement firstcomponent secondcomponent return heightofscc firstcomponent heightofscc secondcomponent swap firstcomponent secondcomponent heightofscc firstcomponent heightofscc secondcomponent firstcomponent currentscc secondcomponent firstcomponent minlevelindfs firstcomponent min minlevelindfs firstcomponent minlevelindfs secondcomponent dfs method void solveit int node int currentlevel int componentofnode getcomponent node minlevelindfs componentofnode min minlevelindfs componentofnode currentlevel visited node true auto currentneighbour graph node visited currentneighbour false solveit currentneighbour currentlevel int componentofneighbour getcomponent currentneighbour minlevelindfs componentofneighbour currentlevel unitecomponents node currentneighbour componentofnode getcomponent node minlevelindfs componentofnode currentlevel minlevelindfs componentofnode maxlevel main method int main ofstream fout output int numberofnodes numberofedges readit numberofnodes numberofedges prepareit numberofnodes int currentnode currentnode numberofnodes visited currentnode false solveit currentnode int howmanyscc int currentnode currentnode numberofnodes currentnode currentscc currentnode fout howmanyscc int currentindexscc int currentnode currentnode numberofnodes int currentscc getcomponent currentnode buildfinalscc currentscc scc buildfinalscc currentscc currentnode else buildfinalscc currentscc currentindexscc scc buildfinalscc currentscc currentnode int currentscc currentscc howmanyscc auto currentnode scc currentscc fout currentnode fout return declarations begin const int maxn const int maxlevel vector int graph maxn vector int scc maxn int currentscc maxn int heightofscc maxn int minlevelindfs maxn int buildfinalscc maxn bool visited maxn end data structure data structure used keeping disjoint sets first described bernard galler michael fischer allows find set specific element belongs join sets two specific elements complexity log numberofelementsinthesets main idea represent set tree root father whole set two sets joined set higher tree considered dominant one father becomes father sets need find father specific element change fathers elements path root linking root two optimizations generate complexity mentioned information may found following links bibliography dfs traversal search algorithm traversing trees graphs directed undirected version developed french mathematician charles pierre tremaux century makes use stack top considered current node idea iterate neighbors current node one visited yet pushed stack set current node information may found following links bibliography bibliography https https
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apr logic assertional logic zhou abstract logic fol widely regarded one important foundations knowledge representation nevertheless paper argue fol several critical issues purpose instead propose alternative called assertional logic syntactic objects categorized set theoretic constructs including individuals concepts operators kinds knowledge formalized equality assertions first present primitive form assertional logic uses minimal assumed knowledge constructs show extend definitions special kinds knowledge assertions argue assertional logic although simpler expressive extensible fol case study show assertional logic used unify logic probability building blocks introduction classical logic fol widely regarded one important foundations symbolic fol plays central role field knowledge representation reasoning many fragments propositional logic modal epistemic logic description logics extensions logic situation calculus probabilistic logic variants datalog answer set programming extensively studied literature nevertheless researchers pointed several issues regarding using fol knowledge representation reasoning mostly reasoning point view first fol computationally difficult reasoning fol undecidable problem also fol monotonic sense adding new knowledge knowledge base always results consequences however human reasoning sometimes nonmonotonic paper argue fol also critical issues knowledge representation point view first although fol considered natural welltrained logicians simple flexible enough knowledge engineers less training one reason distinction hierarchy term level predicate level formula level experience teacher subject although strongly emphasized classes many students failed understand predicate formula scope function another reason notion free occurrences variables instance easily understandable many students gen inference rule enforce variable occurrence restrictions last least arbitrary nesting also raises issues although natural mathematical point view nested formula hard understood used secondly fol limitations terms expressive power fol quantify addressed extending fol logic nevertheless logic still quantify formulas consequence fol logic able represent axiom inference rule logic modus ponens flexible quantification beyond term level needed applications example automated solving mathematical problems often use proof induction represent need state statement number parameter holds numbers less implies holds number well holds natural numbers statement formula level possibly complex within hence order represent use proof induction need quantify formula level thirdly fol hard extended new building blocks fol formalize important notions including probability actions time needed wide range applications purpose researchers made significant progresses extending fol notions separately probabilistic logic situation calculus ctl etc challenging task sense completely syntax well semantics however combing notions together even several seems extremely difficult task moreover many building blocks incorporated applications instance consider task planning home service robots necessary represent actions probability time building blocks preferences altogether time address issues propose assertional logic syntactic objects categorized set theoretic constructs including individuals concepts operators kinds knowledge uniformly formalized equality assertions form either atomic individuals compound individuals semantically individuals concepts operators interpreted elements sets functions respectively set theory knowledge form means two individuals referring element first present primitive form assertional logic uses minimal assumed knowledge primitive constructs show extend building blocks definitions special kinds knowledge assertions used define new individuals concepts operators new syntactic objects defined used basis define example show define using cartesian product nested assertions using show assertional logic although simpler expressive extensible fol case study show extend assertional logic unifying logic probability important building blocks including time note intention reinvent wheel building blocks borrow existing excellent work formalizing building blocks separately assemble within one framework assertional logic live happily ever meta language prior knowledge one build something nothing hence order establish assertional logic need basic knowledge course purpose explanation need informal meta language whose syntax semantics usual use natural language english nevertheless meta language used merely explanation affect syntax well semantics anything defined formally meta level explanation language enough also need core objects knowledge whose syntax semantics well called prior objects prior knowledge instance defining real numbers need prior knowledge natural numbers defining probability need prior knowledge real numbers assertional logic always treat equality symbol prior object prior knowledge associated equality symbol instance equivalence relation satisfying reflexivity symmetricity transitivity also satisfies general substitution property used replace anywhere equality symbol also assume prior objects associated prior knowledge set theory including set operators set union cartesian product boolean values set builder notations natural numbers assertional logic primitive form section present primitive form assertional logic goal assertional logic syntactically represent knowledge application domains two essential tasks capture syntax domain represent knowledge capturing syntax given application domain syntactic structure structure short clear context domain triple collection individuals representing objects domain collection concepts representing groups objects sharing something common collection operators representing relationships connections among individuals concepts concepts operators nested considered individuals well needed concepts concepts concepts operators concepts concepts operators operator could maps tuple individuals single one operator associated domain form representing possible values operator operate concept call arity tuple matching domain operator maps individual denoted operators similar functions fol differs two essential ways first operators could different concepts importantly could constructs concepts concepts concepts operators representing knowledge let syntactic structure term individual either atomic individual result operator operating individuals also call latter compound individuals assertion form two terms intuitively assertion form piece knowledge application domain claiming left right side refer object knowledge base set assertions terms assertions considered individuals well similar concepts group individuals use schemas group terms assertions schema term either atomic concept form concepts essentially schema term represents set terms every concept grounded corresponding individual collection individuals schema assertion form form except terms replaced schema terms similarly schema assertion represents set assertions say schema mentions set concepts occur mentions contains concepts mentioned note could case two different individuals referring concept schema terms assertions case need use different copies denoted distinguish instance assertions human captured schema assertion side schema copy concept refer individual instance human human set assertions form human semantics propose set theoretic semantics assertional logic since assume set theory prior knowledge semantics freely use individuals empty set concepts set natural numbers operators set union operator without explanation interpretation also called possible world pair domain elements mapping function admits prior knowledge maps individual domain element concept set operator function mapping function generalized terms mapping similar terms assertions interpretations also considered individuals studied important emphasize interpretation admit prior knowledge instance since assume set theory suppose interpretation maps two individuals element domain concepts must interpreted must interpreted let interpretation assertion say model denoted iff also written let knowledge base say model denoted iff model assertions say assertion property denoted iff models also models particular say assertion tautology iff modeled interpretations since assume set theory prior knowledge directly borrow set theoretic constructs instance use also written denote new concept unions two concepts applying assertions see assertions primitive form indeed represent many important features knowledge representation instance membership assertion stating individual instance concept following assertion also written containment assertion stating concept contained another concept following assertion also written range declaration stating range operator operating concepts equals another concept following assertion extensibility via definitions argued introduction section extensibility critical issue knowledge representation assertional logic use definitions purpose definitions schema assertions used define new syntactic objects including individuals concepts operators based existing ones new syntactic objects defined used define note definitions nothing extra special kinds knowledge assertions start defining new individuals individual definition assertion form atomic individual term individual defined assertion claims left side defined right side instance means individual defined empty set defining new operators similar defining new individuals except use schema assertions instead let operator defined domain operator definition schema assertion form schema term mentions concepts since schema assertion represents set assertions essentially operator definition form defines operator defining value instance defining successor operator succ use schema assertion succ meaning every natural number successor defined defining new concepts somewhat different concepts essentially sets directly borrow set theory notations purpose four ways define new concept enumeration let individuals collection concept written instance define concept digits digits operation let two concepts union intersection difference cartesian product power set concepts operation written assertions well instance following assertion states concept defined union example one define concept human ale restricted comprehension let concept schema assertion mentions concept individuals satisfying denoted simply form concept written instance define concept ale ale animal animal male meaning ale consists animals whose sexes male replacement let operator concept well defined individuals mapped denoted simply form concept written instance define concept arents arents arentof human meaning consists individuals arentof human definitions incremental may define syntactic objects first defined used define one always continue incremental process instance arithmetic define successor operator first defined used define add operator served basis define since terms assertions considered individuals define new type terms assertions definitions example extend assertions form using cartesian product first define fixed number assertions given number define new operator arity following schema assertion concepts terms notice single assertion form sense multiassertion syntax sugar define concept ulti assertion copies standard assertions convenience use assertionn denote defined used define syntactic objects example use define nested assertions first define nested terms follows ested erm erm erm erm ested erm nested assertions defined ested assertion ested erm ested erm nested assertion defined used basis define forth using nested assertions simplify representation task however one overuse since essentially every use nested term introduces new individual embedding fol assertional logic previous section show extend assertions primitive form nested assertions section continue task show define complex forms assertions logic connectives including propositional connectives quantifiers start propositional case let concept nested assertions introduce number operators assertional logic including negation conjunction disjunction implication could different ways define operators assertional logic let two nested assertions propositional connectives defined follows used also denote one observe ranges logic operators nested assertions hence similar nested assertion propositional logic operators syntactic sugar well consider define operators quantifiers including universal quantifier existential quantifier domain quantifiers pair concept schema assertion mentions quantifiers defines follows intuitively true iff individuals holds equals concept individuals holds true iff individuals holds equal empty set exists least one individual holds see ranges quantifiers nested assertions well sense quantifiers also syntactic sugar primitive form note quantifiers defined ranging arbitrary concept concept atomic individuals quantifiers range concept quantifiers nevertheless concepts could different case fol moreover could complex concepts concept possible concepts case monadic logic yet could many concept assertions concept concepts terms etc sense quantifiers become finally biggest difference even concept assertions quantifiers assertional logic quantify assertions corresponding formulas classical logics done classical logics including logic verified tautologies propositional logic fol demorgan laws also tautologies assertional logic space reasons leave theorems proofs full version paper incorporating probability probability another important building block knowledge representation last several decades development uncertainty artificial intelligence number influential approaches developed important applications found machine learning natural language processing etc normally incorporate probability logic one complete redefine whole semantics since integrations probability logic connectives quantifiers complicated section show easily done assertional logic key point although interactions logic probability complicated interactions assertions basic form relatively simple shown previous section interactions logic assertions defined lines section following gaifman idea show indeed case integrating assertions probability well interactions logic probability automatically established via assertions integrating assertions probability since operations real numbers involved defining probability need assume theory real number prior knowledge gaifman proposed define probability logic sentence sum probabilities possible worlds satisfying following idea assertional logic introduce operator probability concept assertions range concept real numbers possible world assign associated weight positive real number assertion probability denoted define following schema assertion definition defines interactions probability assertions case number infinite worlds need use measure theory nevertheless beyond scope paper defined probability assertion real number directly use inside assertions sense valid assertions able vefiry properties probability instance kolmogorov first second probability axioms also extend definition conditional probability introduce new operator pairs two assertions following similar idea conditional probability assertion providing another assertion also denoted defined following schema assertion conditional probability defined real number use arbitrarily inside assertions similarly verify properties conditional probabilities including famous bayes theorem assertions interactions logic probability via assertions although define probabilities assertions basic form interactions probability building blocks logic automatically established since assertions connected logic operators reduced primitive form sense investigate properties interactions logic probability instance observed kolmogorov third probability axiom tautology assertion logic let assertions pairwise disjoint verified many axioms properties regarding interactions logic probability tautologies assertional logic instance additivity axiom distributivity axiom implies two assertions sense assertional logic also used validate existing properties interactions logic probability addition may foster new discoveries interactions logic probability properties nested probabilities note intend reinvent wheel defining probability interactions logic definitions conditional probability borrowed literature instead take probability case study show one building block logic another probability interacted assertions without going deeper interactions building blocks critically many important building blocks incorporated barely possible clarify interactions among nevertheless becomes possible unify altogether assertional logic one needs consider interactions building blocks basic form assertions separately consequently interactions among building blocks automatically established via assertions unifying logic probability another case study consider formalize time assertional logic time understood different ways time points time interval ltl ctl following idea incorporating probability need consider interactions time assertions paper report simple case integrating assertions time points let concept time points introduce new operator whose domain pair intuitively value individual time point introduce temporal formulas new boolean operator whose domain pair following schema assertion interactions time points logic connectives probability automatically established able investigate properties instance assertions iff etc hence integrated formalism combing logic probability time points assertional logic discussion related work conclusion paper argue purpose knowledge representation classical firstorder logic critical issues including simplicity flexibility expressivity extensibility address issues propose assertional logic instead syntax application domain captured individuals objects domain concepts groups objects sharing something common operators connections relationships among objects knowledge domain simply captured equality assertions form terms assertional logic without redefining semantics one extend current system new syntactic objects definitions special kinds knowledge assertions defined syntactic objects used define done assertional logic extend primitive form assertional logic nested assertions well logic connectives quantifiers consider extend assertional logic important building blocks key point one wants integrate new building block assertional logic needs formalize syntactic objects including individuals concepts operators defines interactions basic form assertions interactions building block others automatically established since complicated assertions essentially reduced basic form case study briefly discuss incorporate probability time points paper course assertional logic deeply rooted logic individuals concepts operators analogous constants unary predicates functions respectively assertions originated equality atoms nevertheless differ many essential aspects firstly individuals objects concepts assertions concepts operators secondly assertional logic naturally domain operator tuple many different concepts including ones thirdly concepts play central role assertional logic natural human knowledge representation concepts formalized unary predicates fol specifically emphasized fourthly assertional logic kinds knowledge uniformly formalized form equality assertions shown section complicated logic sentences defined equality assertions well embedding connectives quantifiers operators assertions fifthly following although connectives quantifiers nesting defined assertional logic considered primitive constructs sense used demand necessary argue important reason makes assertional logic simpler fol sixthly assertional logic simple form expressive constructs inherently related within rich syntactic structure contrast equality atoms fol power last least assertional logic directly embraces extensibility within framework syntactic definitions instance define quantifiers assertional logic needs two lines see equations without redefining whole new semantics much simpler fol assertional logic also inspired many approaches symbolic traditionally fol strict hierarchy term level formula level extent broken situation calculus game description language quantify situations actions fluents directly talk whether fluent holds particular situation hold assertional logic goes much completely demolishing distinction hierarchy term level predicate level formula level assertional logic one flexibly use atoms predicates scope function long match domain definition also one quantify concepts including concept assertions makes assertional logic even expressive logic quantifier formulas another inspiration comes family description logics terminologies individuals concepts borrowed family description logics allows certain level extensibility interpreting individuals concepts roles domain elements unary predicates binary predicates respectively one extend basic description logics building blocks nominal number restrictions inverse transitive roles etc based foundational semantics also one freely assemble building blocks different fragments description logics alc shiq shion however many important building blocks actions probability time rules etc still difficult incorporated method interesting pioneering work done consider extensibility description logics nevertheless differ assertional logic embraces extensibility syntactic level instead semantic one paper concerned representation task definition task leave reasoning task future work certainly complete reasoning assertional logic undecidable fol embedded traditionally way address issue find decidable fragments nevertheless shall propose alternative approach focuses efficient necessarily complete reasoning developed approach flexibility extensibility assertional logic play critical role shall present another paper nevertheless argue representation definition worth study merits successful stories include diagram semantic network many besides extending assertional logic important building blocks actions effects indeed challenging worth pursuing acknowledgement author gratefully acknowledges fangzhen lin comments first draft paper references james allen maintaining knowledge temporal intervals commun acm november franz baader diego calvanese deborah mcguinness daniele nardi peter editors description logic handbook theory implementation applications cambridge university press new york usa franz baader philipp hanschke scheme integrating concrete domains concept languages proceedings international joint conference artificial intelligence volume ijcai pages san francisco usa morgan kaufmann publishers fahiem bacchus representing reasoning probabilistic knowledge logical approach probabilities mit press cambridge usa alexander borgida extensible knowledge representation case description reasoners artif intell res jair ronald brachman hector levesque knowledge representation reasoning elsevier edmund clarke allen emerson design synthesis synchronization skeletons using temporal logic logic programs workshop pages london haim gaifman concerning measures first order calculi israel giuseppe giacomo maurizio lenzerini riccardo rosati description logics domain metamodeling proceedings aaai conference artificial intelligence aaai pages aaai press theodore hailperin probability logic notre dame formal logic paul halmos naive set theory van nostrand reprinted springerverlag undergraduate texts mathematics joseph halpern analysis logics probability artif thomas keller patrick eyerich bernhard nebel task planning autonomous service robot proceedings annual german conference advances artificial intelligence pages berlin heidelberg kutz lutz wolter zakharyaschev abstract description systems artificial intelligence hector levesque fiora pirri ray reiter foundations situation calculus electronic transactions artificial intelligence vol issue fangzhen lin situation calculus handbook knowledge representation pages brian christopher milch probabilistic models unknown objects phd thesis berkeley usa judea pearl probabilistic reasoning intelligent systems networks plausible inference morgan kaufmann publishers san francisco usa amir pnueli temporal logic programs proceedings annual symposium foundations computer science sfcs pages washington usa ieee computer society matthew richardson pedro domingos markov logic networks machine learning michael thielscher proposal extend game description language general epistemic games ecai european conference artificial intelligence september hague netherlands including prestigious applications artificial intelligence pais pages frank van harmelen vladimir lifschitz bruce porter editors handbook knowledge representation volume foundations artificial intelligence elsevier
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two distinct seasonally fractionally differenced periodic processes mar ahmed bensalma enssea universitaire tipaza inps doudou mokhtar benaknoun algeria faculty mathematics university science technology houari boumediene algeria abstract article devoted study effects fractional differencing filter put effect evidence derived periodic functions two distinct univariate seasonally fractionally differenced periodic models multivariate representation periodically correlated process exploited provide exact approximated expression models distinction models clearly obvious expression periodic function besides producing different functions two models differ implications first model seasons multivariate series separately fractionally integrated second model however seasons univariate series fractionally simulated sample models parameters empirical periodic autocovariance calculated graphically represented illustrating results support comparison two models introduction since introduction gladyshev much attention given periodically correlated processes interest processes due potential use modeling cyclical phenomena appearing hydrology climatology econometrics following pioneer work gladyshev important part literature devoted periodically correlated discrete time processes discrete time process periodically correlated non zero integer cov cov review periodically correlated discrete time processes proposed lund basawa bentarzi hallin give invertibility conditions periodic moving average large part literature subject devoted periodic arm arm models following representation white noise variance among searchers interested periodic autoregressifs processes periodically stationary cite boswijk franses studied problem presence unit root periodic autoregression model order boswijk franses haldrup studied presence multiple unit roots periodic autoregression model order work cited made assumption processes periodically integrated order zero integrated order one periodically integrated order one however currently scientific fields mentioned hydrology meteorology econometrics much sets data certain periodicity also long range dependence long memory phenomena modeled stationary processes stationary processes seasonal long memory well know see example gray zhang woodward garma models seasonal arf oppenheim ould haye references properties simulations another alternative take account certain periodic phenomena long memory consider nonstationary models periodically stationary periodically correlated processes long memory periodically correlated processes within meaning gladyshev long memory receive much attention behalf statisticians probabilists among works associating periodicity within meaning gladyshev presence long memory cite hui franses ooms ooms franses modelling hong kong united christian hospital attendance series hui propose correlated process zero mean white noise variance fractional parameter empirical series concerns seventy five approximately one half years data average number people entering emergency unit weekday weekend hand order analyzes properties conditional mean quarterly inflation rate united kingdom franses ooms propose correlated process defined fractional parameter finally monthly empirical data concern log transformed data monthly mean river flow cubic feet per second ooms franses propose use seasonal periodic fractional operator defined simple framework follows defined main difference one hand models hand model unit lag fractional difference operator applied models fractional difference operator applied weekly quarterly lags respectively corresponding basic time interval time series analyzed model fractional difference operator applied yearly seasonal lag time series analyzed indeed using binomial expansion difference operator rewrite respectively models following defined terms expressions recurrence formula invertibility stationarity conditions model known see ooms franses apart constant nothing clear models precisely thing clear stationarity conditions model infinite moving average representation unknown thing clear invertibility conditions model infinite autoregressive representation unknown model invertible stationary easy show case infinite autoregressive representation process given model case general model case general particular periodic arf namely infinite moving average representation unknown paper give closed form representation important known representation order deduce stationarity condition type model unfortunately closed form obtained easy handle due parametric complexity see appendix since arf easy handle work present article concerned seasonal periodical fractional operator namely precisely work interested certain theoretical properties arf seasonal periodic arf study theoretical properties class models remains made among works evoke class one exists franses ooms work franses ooms consist adjusting arf set real data precisely model considered ooms franses defined follows constant parameters periodic functions white noise seasonally fractionally integrated order fractional parameter model written follows odel another class models arf distinct used franses ooms class defined follows odel defined like two classes coincide since generally composition necessarily commutative convince sufficient notice representation model arf model vector autoregressive model driven fractionally integrated innovation whereas multivariate writing model model fractionally integrated vector autoregression see rebecca sela clifford hurvich two distinct classes generalize univariate model arf first closely related cointegrated processes whereas second closely related integrated processes consequently case model closely related cointegrated season model closely related integrated season order distinguish model note respectively following rest paper organized follows section devoted defined two class processes periodic autoregressive order process periodic seasonal fractional integrated order innovation namely periodic seasonal fractional integrated process periodic autoregressive order namely section model defined section provide exact approximated expression periodic autocovariances function section simulated samples model parameters model empirical periodic autocovariances calculated graphically represented illustrating theoretical results comparison two models without restricting generality suppose processes defined zero mean representation notation seasonally fractionally integrated periodic autoregressive process periodically correlated process said seasonally fractionally integrated order periodic autoregressive order following representation zero mean white noise variance defined like parameters letting process rewritten variate form denotes smallest integer equal defined like autoregressive coefficient matrices given periodic stationarity condition model stationarity condition equivalent fractional integrated vector autoregression namely rebecca sela clifford hurvich representation means roots determinantal equation det less absolute value hannan fuller hosking process stationary infinite moving average representation given diag sequence absolutely summable matrix defined like ith element written follows ith rows see clearly integrated order periodic autoregressive seasonally fractionally integrated process periodically correlated process said periodic autoregressive order seasonally fractionally integrated order following representation defined like letting process rewritten variate form denotes smallest integer equal defined like model vector autoregression fractional integrated innovation namely arf rebecca sela clifford hurvich periodic stationarity condition model model ith relation written ith rows means ith relation integrated order among relations integrated order lower max relations cointegration values different say relations cointegrations exist relation cointegration generally means relations cointegrations seasons model stationary infinite moving average representation given ith element written follows element matrix written like linear combination independent processes respectively integrated order consequently integrated order max granger periodic autocovariances section deals determination theoretical periodic autocovariances periodically correlated processes defined precedent section periodic autocovariances theorem given stationary process defined element matrix ith rows matrix diag proof see chung corollary given process defined integers defined follows sth rows matrix diag proof proof corollary rises directly theorem theorem cov element covariance matrix moreover known cov cov putting replacing cov according value equality becomes using approximation sth rows matrix diag corollary emerges several remarks important remark periodic autocovariances taper different hyperbolic rates suppose min max restrict generality speedy taper hyperbolic rate lowest taper hyperbolic rate remark largely clarified graphically see section couples figures advantage offer periodic process possibility representing graph autocovariances various manners autocovariances functions represented plot hui separately also represented plot three kinds graphs use next section periodic autocovariances stating main result section need notation let dmax max define stating main result section need notation let dmax max fine dmax dmax theorem given stationary process defined element matrix dmax dmax element proof see chung corollary gives approximated expression periodic autocovariances function cov process defined corollary given process defined dmax dmax integers defined follows element matrix proof proof corollary rises directly theorem theorem dmax dmax cov element covariance matrix moreover known cov cov putting replacing cov according value equality becomes using approximation max max dmax remark periodic autocovariances model coincide model remark corollary see periodic autocovariances taper hyperbolic rates simulation section compare finite sample periodic autocovariances models different value sample size model model consider simulation study model following representation model following representation model following representation diag simulated autocovariances model figures represent empirical autocovariances function plot model different value figure periodic autocovariances lag model figure periodic autocovariances lag model figure periodic autocovariances lag model figure periodic autocovariances lag model figures illustrate well theoretical result theorem also states periodicity caused fractional parameters lag taper respectively hyperbolic rates according value simulated autocovariances model figures represents empirical autocovariances respectively spike graph line graph model couples figures represents empirical autocovariances respectively spike graph line graph model figure figure periodic autocovariances lag model taper different hyperbolic rates figure figure figure represents respectively speedy lowest taper hyperbolic rate autocovariances model figure figure periodic autocovariances fixed tendency increase according value figure figure periodic autocovariances fixed tendency increase according value figure figure periodic autocovariances fixed tendency increase according value figure figure periodic autocovariances fixed tendency increase according value simulated autocovariances model figures represents empirical autocovariances respectively spike graph line graph model difference periodic autocovariances lag decreases manner mainly taper hyperbolic rates figure figure periodic autocovariances lag taper hyperbolic rates simulated comparison autocovariances model order compare autocovariances model model represent graphically scale different value see figure periodic autocovariances lag respectively model model figure periodic autocovariances lag respectively model model figure periodic autocovariances lag respectively model model figure periodic autocovariances lag respectively model model figures plot autocovariance sequences model model scale identical parameters autocovariances sequences differ dramatically rebecca sela clifford hurvich presents similar conclusion sequences bivariate arf processes parameters point first model series integrated separately case seasons integrated separately second cointegration relation two series case cointegrations relations four seasons fact explain clearly difference autocovariances model model taper hyperbolic rates autocovariances model equal lowest tapper hyperbolic rate autocovariances model autocovariance sequences differ dramatically explanation explicit results corollary corollary generally literature long memory models attention focused fractional parameters associate hyperbolic tapper autocovariance rather autoregressive moving average parameters included expression autocovariance expression autoregressive parameters appears following form expression appears following form model model set possible values two quantities respectively seen possible values greater greater see diagonal matrix hand values lower except last value diagonal matrix conclusion model allowing seasonal fractional parameter rather constant highlighted existence two distinct models see section model model two distinct models established exact approximated expression periodic autocovariance simulated sample model empirical periodic autocovariance calculated graphically represented clear theoretical simulated results easy distinguish two models shape autocovariance model sufficient consider general model namely situation becomes complex handle number different models distinguish two models furthermore non seasonal part general model arf receive much attention behalf statisticians probabilists references peter boswijk franses philip hans testing periodic integration economics letters elsevier vol pages boswijk peter franses philip hans haldrup niels multiple unit roots periodic autoregression journal econometrics elsevier vol pages chung sample means sample autocovariances linear regression stationary multivariate long memory processes econometric theory franses ooms periodic long memory model quartely inflation international journal forecasting gladyshev periodically correlated random sequences soviet mathematics gladyshev periodically almost random processes continuous time parameter theory probability applications gray zhang woodward generalized fractional processes time ser granger long memory relationships aggregation dynamic models journal econometrics granger developements study cointegrated economic variables oxford bulletin economics statistics hosking fractional differencing biometrica hui fractionally differenced periodic processes sankhya ser lund basawa modeling inference periodically correlated time series asymptotics nonparametrics time series volume statist textbooks pages dekker new york rebecca sela clifford hurvich computationally efficient methods two multivariate fractionally integrated models journal time series analysis wiley blackwell vol pages ooms franses seasonal periodic long memory model monthly river flows envoronmental modelling software volume issue pages oppenheim ould haye viano long memory seasonal effects statist inf stoch ould haye viano limit theorems seasonal longmemory doukhan paul theory applications dependence boston birkhuser porter hudak aplication seasonal fractionally erenced model monetary aggegrates journal american statistical association appendix proposition infinite moving average representation process defined given numwhere ber terms sum equal number represent cardinal sets positive integers namely summed together give proof putting rewrite generally suppose infinite moving average representation given lagged variable replacing obtain putting becomes let rewrite rewrite let rewrite infinite moving average representation process identification obtain first three coefficients generally constant
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analyzing attacks cooperative adaptive cruise control cacc rens van der heijden thomas lukaseder frank kargl dec institute distributed systems ulm university germany adaptive cruise control cacc one driving applications vehicular networks vanets promises bring efficient faster transportation cooperative behavior vehicles cacc vehicles exchange information relied partially automate driving however reliance cooperation requires resilience attacks forms misbehavior paper propose rigorous attacker model evaluation framework resilience quantifying attack impact providing necessary tools compare controller resilience attack effectiveness simultaneously although significant differences resilience three analyzed controllers show attacked effectively easily either jamming data injection results suggest combination misbehavior detection resilient control algorithms graceful degradation necessary ingredients secure safe platoons ntroduction paper study cooperative adaptive cruise control cacc application vehicular networks vanets aimed increasing road efficiency partially automating driving cacc essentially extension existing cruise control adaptive cruise control acc technologies designed allow driver maintain constant speed driving case acc vehicular sensors radar lidar cameras used measure distance preceding vehicle order automatically respond changes driving behavior cacc extends concept allowing vehicles communicate create platoon consisting leader vehicle multiple followers shown figure many different controller implementations literature enable behavior since proposal cacc proposal shown cacc efficient acc even constant spacing vehicles independent platoon speed theoretically achieved whereas shown acc alone achieve goal since cacc proposed vanets made significant leaps development including dsrc standard variety etsi ieee sae standards define various network layers parallel developments security standards designed order protect attacks common fields relatively early research ieee accepted vnc doi available yet victims attacker leader fig figure shows platoon leader several followers work red vehicle attacker platoon length attacker transmits malicious beacon false acceleration became clear integrity significant security goal vanets general although availability also significant ieee standard defines protect common external attacks providing message integrity message authenticity legitimate vehicles transmit messages europe camp since started deployment security credential management systems scmss manage constitutes legitimate vehicles parallel one major advances security community introduction misbehavior detection mean detect revoke vehicles transmit malicious data type behavior focus work particularly desired increased road usage cacc requires vehicles move way would unstable acc deployed research misbehavior detection vanets concentrates mostly able detect incorrect information beacon messages messages defined previously mentioned standards periodically sent neighboring vehicles inform state sender one main application areas studied evaluate detection schemes routing communication enabled finding routes based vehicle positions vehicle position also used order avoid potential accidents normal traffic warning driver alarm tone many detection algorithms operate position information specifically however cacc typically uses acceleration speed information compute behavior local vehicle based leader predecessor vehicles cases entire platoon information becomes unavailable incorrect accidents may occur data cacc controllers essential protect security controllers considering malicious insiders studied authors existing work shows attacks impact safety potentially cause accidents lack rigorous analysis framework within attacks resilience respective controllers analyzed paper lay groundwork framework implement several attacks provide concrete quantification measure impact attack going beyond simply finding whether accident occurs analyze significance attack terms accident quantify stability individual platoon affected even accident occurs present results three existing controllers implemented plexe use extensible platooning simulator basis evaluation work easily extended new controllers similarly attack implementation easily extended study impact attacks publish source code data analysis scripts recorded data order stimulate research area remainder paper organized follows section discusses existing analyses attacks cacc platooning well basics regarding controllers plexe section iii divide existing attacks three categories two analyzed extensively paper also discuss proposed attacks infeasible assumption current security standards implemented correctly section describes simulation setup corresponding evaluation discuss results implications cacc section close conclusion section elated ork distinguish two areas related work controllers implement functionality cacc acc offer existing studies misbehavior platooning similar settings controllers platoons vehicles driving group attempting keep minimal still safe distance vehicles within group achieved using acc cacc main reason using cacc safe distance much smaller settings platoon normally consists single leader set followers together referred platoon sometimes string set followers dynamic although usually requires kind join leave procedures vehicle implements controller determines desired acceleration used configure vehicles engine brake controller work focus upper controller setting desired acceleration exclusively since controller attacker easily reach either communication attacks sensors broad scope literature control theory engineering side covering variety different controllers literature typically introduces two stability metrics local stability string stability local stability refers vehicles distance next vehicle string stability property platoon determines spacing errors propagate platoon platoon considered string stable transfer function less basically means platoon eventually converge stable state leader keeps specific velocity perfectly maintaining distance next vehicle done keeping fixed distance meters maintaining headway time seconds referred constant spacing spacing earlier work shown noncooperative methods acc achieve string stability constant spacing policy work focus controllers implemented segata plexe includes constant spacing controller controller graceful degradation consensus controller constant spacing controller taken rajamani keeps fixed distance based measured distance received speed acceleration preceding vehicle leader vehicle controller ploeg designed enable graceful degradation network errors degraded cacc controller predicts acceleration preceding vehicle information received idea degraded controller outperforms acc thus used bridge communication gaps particularly interesting jamming attacks last controller consensus algorithm bernardo uses position speed acceleration information vehicles platoon controller larger spacing aims maintain stability bidirectionally vehicles also adapt behavior vehicles behind particularly interesting respect data injection attacks existing attacks attacks cacc typically studied two different perspectives mostly working parallel security research vehicular networks already suggested new types attacks overall network previous decade recent work concentrated misbehavior detection data injection attacks uniquely suitable vehicular networks cyberphysical systems platooning one motivating examples researchers hand authors control theory side gained understanding potential security concerns started developing control algorithms terms attacks authors also investigated exploit knowledge controller show attacks causing instability possible discussed platoon behavior jamming attack scenario jammer drone flying platoon aiming use limited power disrupt platoon jamming thus done reactively success rate conserve power although power consumption considered work shows acc controller string stable seconds headway distance cacc influence reactive jammer string stable seconds headway distance additionally show attack works best attacker located near first vehicle platoon amoozadeh discuss controller model describe variety attacks including application network layer attacks well issues sensor tampering privacy issues present result message falsification attack setting vehicle external platoon falsifies beacons result attack platoon instability platoon converge back stable state also explain attack effective acceleration changes occur similar results presented jamming attack authors controller downgrades acc automatically work aim show downgrade necessarily sufficient depending system parameters chosen debruhl also use two component controller consisting feedback controller keep distance feedforward controller whose outputs added together form desired acceleration provided vehicle drivetrain controller shown real networked platoons debruhl examine behavior various attack strategies including collision induction attack authors continue develop error calculation detection algorithm essentially estimates expected behavior vehicle front switches acc vehicle appears behave differently claims outside context communication also relevant secure safe platooning work research group gerdes dadras authors looked platoons examined adversarial behavior within platoon attacker systematically disrupts distributed control algorithm using another algorithm manipulates control input vehicle achieve goal first work authors show given pid controller cause significantly increased energy consumption neighboring vehicles reduced energy consumption one main drivers cacc research impact significant second work shows controller attacker cause platoon asymptotically unstable destabilize eventually dissolve platoon iii attacker odel section describe attacker model covers attacks proposed literature well elements area misbehavior detection attacks sensors particular interested realistic attacks proposed literature considering currently standardized security mechanisms standardized security mechanisms include use pseudonymous certificates protect driver privacy corresponding private keys stored hardware security module hsm somewhat protected tampering core attacker model attacker may gain control vehicle thus legitimate key material happen either software compromise malware spread applications running entertainment system physical manipulation vehicle key extraction old hsm however unlike traditional network attacker models attacker additionally must obey laws physics observable network participant attacker omnipresent throughout network cacc specifically assume software protocol implementations suitable attacker find exploit must obey protocol words restrict target attack platooning application assume mechanisms place filter invalid improperly formatted messages model attackers primary goal cause crash within platoon secondary goal destabilizing assume attacker already part platoon position attacks damage however literature points attacker controls leader vehicle complete control platoon therefore attacks performed vehicle follow argument also assume attacks executed follower order study potential impact attacks another argument leader vehicles placed additional scrutiny replaced necessary followers difficult replace jamming denial service dos attacks includes variety jamming attacks regular reactive popular variant attack within vehicular networks also includes various attacks higher layers influence reception violating mac protocol causing others wait authors also suggest dos attack may launched network layer order disrupt communication overloading hardware security module hsm leads form cryptographic packet loss packet loss caused messages verified verified time work consider attacks effect messages longer arrive refer jamming generalizes attack compared related work authors considered reactive jamming data injection second class attacks identify data injection generalizes variety different attacks discussed related work attack requires attacker vehicle compromised attacker send modified packets falls category particularly useful types attacks may feasible mitigate misbehavior detection specific implementation attack usually dependent type data exchanged controller thus corresponding impact different every controller similarly goal attacker significant implementation sometimes used distinguishing factor literature includes reduced headway attack collision induction attack misreporting attack among others however goals often achieved one classes attacks collision induction possible jamming thus avoid terminology spoofing replay attacks also considered examples class considered mostly solved vehicular networking security community vehicular networks gps often assumed practically ubiquitous replay attacks protected using close synchronized clocks similarly cryptographic keys authenticated pseudonymous certificates messages protected digital signatures spoofing attacks cause receiver misinterpret origin message possible also applies message falsification attacks attacks manipulate messages vehicles digital signatures adequate countermeasure however possible use multiple certificates simulate one vehicle referred sybil attack originally introduced douceur whether attack feasible depends easy attacker obtain pseudonymous certificates used simultaneously sensor manipulation third final class attacks sensor manipulation constitutes attack causes internal network vehicle report incorrect information controller attacker could either access internal network vehicle transmit false data causing vehicle react incorrectly even destabilize platoon controller attack although could also done software manipulating memory controller processes inputs attacks internal network considered interesting misbehavior detection potential countermeasure software attacks hand difficult prevent misbehavior detection detection system likely also compromised setting another variant sensor manipulation positions attacker outside target vehicle instead attacking network attacker uses sensors tools blind sensors target vehicle cause behave certain way researchers shown lidar cameras vulnerable attacks effect attack similar type sensor manipulation attack controller sees invalid inputs distinction attacker located influences cost attack expected results platoon behavior similar thus study attacks place attacks category table xcerpt simulation parameters full parameters available data source code repositories transmit power sensitivity platoon length attacker controllers cacc spacing target speed attack speed value attack accel value attack position shift cacc ploeg consensus attack nalysis paper focus communication want answer question cacc behaves misbehavior potential impacts attacks aim work provide general framework attack analysis helpful development resilient control algorithms well misbehavior detection community simulation setup use plexe version simulation toolkit based veins basis analysis due flexibility implementing control algorithms accurate simulation network behavior plexe also extends sumo provides realistic microsimulations physical vehicles extends toolkit various controllers constant spacing controller controller consensus controller implementation leave plexe mostly unchanged extend include attacks implement four attacks one jamming attack three data injection attacks jamming attack designed analyze maximum possible impact attack therefore implemented locally dropping received messages attack starts simulates type denial service including cryptographic packet loss various jamming strategies provides upper bound attack success although may overestimate cases alternative would implement specific attack done alipourfanid approach generalizable data injection attacks implemented exchanging data false data message attacker would send according normal control algorithm analyze three different types injecting false positions false speeds false acceleration values multiple ways implement attacks case set attackers value constant value speed acceleration greatly simplifies interpretation results addition aim provide upper bound attack effectiveness position falsification provide position error increases linearly time added vehicles current position beacon simulation scenario analyze standard scenario plexe used authors analyze platoon behavior sinusoidal scenario scenario initializes platoon provides leader specific acceleration profile leads sinusoidal speed graph performing attacks different points time use scenario approximate impact attacks real behavior significant simulation parameters found complete configuration file code available way plexe implemented simulating chain collisions reliably possible collected information first collision metrics work aim analyze effectiveness various attacks different cacc controllers goal attacker either cause crash maximize instability platoon distinguish two cases crash instability crash occurs impact velocity used quantify impact attack medical research suggests best predictive factor injury crash quantify instability platoon remainder simulation approximately seconds since studied controllers shown string stable assume platoon eventually stabilizes option chose maxs simulation run ith vehicle platoon time spacing error accurately represent worst case inspired earlier work similar metric used spacing error defined difference desired distance controller actual distance two vehicles alternative metrics avgs avgi maxt average maximum spacing error avgs avgi maxt average maximum acceleration considered metrics passenger comfort usability studies shown spacing important user acceptance included acceleration additional metric also common transportation research averaging simulations vehicles allow estimate average comfort vehicles platoon however maximum spacing error accurately represents potential inefficiency risk due attack decided https data repository paper includes metrics https results sim tim section presents results attack experiments shows used metrics gain insight effectiveness different attacks well resilience existing cacc algorithms data results based also published although also using source code configuration mentioned fig heat map showing attack success impact constant jamming attack given point simulation time different target platoon speeds different controllers attack leads accident number impact velocity accident number maximum spacing error leader target speed leader target speed loe loe loe loe fig effect position falsification blue areas represent points related crash fig attacking false speeds blue areas represent points related crash jamming first discuss results jamming attack shown figure figure includes successful attacks crashes marked red stability impact crash marked blue well corresponding impact see section figure shows aggregated values using previously discussed aggregation approach five runs cases impact attack remained similar five runs resulted collisions five runs result collisions simulated different target speeds different jamming times target speed intended average speed platoon average speed leader vehicle jamming time time jamming starts jamming time directly corresponds range acceleration leader vehicle positive since leader data disseminated throughout platoon attacker use acceleration profile choose time attack based speed profile generated plexe standard configuration determined positive acceleration lies seconds simulation time since behavior sinusoidal pattern repeats every seconds seen figure target speed impact effectiveness attack cacc however attacks successful higher velocities ploeg consensus used spacing algorithms depends platoon target speed therefore conclude spacing strong influence attack effectiveness hand attack effectiveness directly influenced target velocity general spacing attack impact decreases figure shows always true simulation time due fact lower attack occurs attacker car subsequent car car avoid colliding attacker however car collide car position injection next step analysis covers position falsification attacks often cited one significant types data injection attacks vehicular networks model false position shift attackers real position increases time turns attack completely ineffective ploeg constant spacing controllers causes crashes consensus controller used however furthermore see spacing error ploeg controller increases significantly leader target speed consensus controller one uses received position information directly somewhat surprisingly impact attack consensus controller depends value transmitted attacker significantly target speed leader thus platoon moving full results shown figure shift parameter mark color scatter plot different leader target speeds top left right graph includes results four different controller configurations bottom left right impact speed successful attacks white background varies maximum spacing error gray background unaffected position falsification means attacks effect lead accident speed injection next attack manipulating speed transmitted attacker continuously output value regardless platoon controller behavior causes error attacker value actual value vary falsifies platoon target speed result analysis displayed figure shows speed injection successful constant spacing cacc significant crashes happen lower speeds attacker claims accelerate actually attacker claims negative speeds result high position error ploeg controller shows behavior affected attack consensus controller also unaffected although spacing error much higher due constant falsified speed attacker achieve success constant value matching leaders speed expect change leader behavior less predictable acceleration injection final attack performed manipulation transmitted acceleration also chose relatively extreme scope order discover maximum potential attacks resulting statistics shown figure expected attack affects controllers acceleration used leader target speed loe loe fig behavior controllers falsified acceleration information blue areas represent points related crash control algorithms however effect widely different per controller consensus controller tolerate large errors acceleration information although maximum spacing error increases slightly others fail almost completely likely due fact consensus controller coupled controller precisely vehicle considers others control algorithm similar speed injection attack injecting false acceleration data causes attack impact dependent injected value attack impact significant positive acceleration values however impact false negative acceleration also significant however point distance sensor notices distance large resets acc behavior iscussion results show cacc controllers unreliable subject effective denial service attacks impact injection also significant data used controller directly confirms earlier results individual jamming injection experiments shows results also hold simulation parameters changed beyond results show window opportunity attacker jam transmit false information broad attacker need hit precisely correct timing attack work results also used guide attacker hone effectiveness attack resilient control algorithms overall conclude ploeg controller deals well jamming even considering controller intended intermittent unreliable channels complete failure however still managed cause accidents even algorithm suggesting eventual complete fallback acc may advisable consensus controller behaves differently fails later starting point jamming future work could thoroughly investigate cause behavior finally conclude constant spacing controller completely unsuitable unreliable communication window causing crash least seconds data injection attacks actual value transmitted attacker important influence attack impact target speed leader matter much comparison expected attacks increase impact overall platoon speed increases expected controllers react quickly scenarios although case position errors crash impact remains relatively consistent cacc controller ploeg controller impact dependent speed dependence mainly due gap occur vehicles one exception observations consensus controller surprisingly resilient attacks showing impact speed acceleration falsification however impact position shift controller significant nevertheless suggests consensus controller might good starting point building resilient controller despite relative ineffectiveness illustrated higher overall position error results likely extend similar scenarios cacc ploeg controllers rely leader preceding vehicle platoon long enough contain crashing vehicle affected potential crashes unclear well results transfer consensus controller since intended behavior quite different two controllers considering vehicles platoon control vehicle however overall methodology applied constrained scenario tools provide easily used tune parameters controllers increased safety attackers parameters improve attack effectiveness onclusion uture ork paper discussed holistic attacker model implemented several attacks tested impact existing controllers using proposed analysis methodology overall results show controllers vulnerable attack significant differences controllers terms affected attacks research community critically discuss limitations cacc influence attacks future work focuses improving attacks studying detection however detection alone likely insufficient volume crashes encountered work therefore focus also prevention design resilient controllers one hand effective detection revocation mechanisms future work false data injection could look types controllers behave study effectiveness simultaneously falsifying multiple fields beacon latter particularly relevant circumvents many typical misbehavior detection mechanisms could detect attacks described reasonable accuracy part due values chose alternative route yet studied well injection false data external sensors research shown camera lidar sensors tricked easily lab environment plexe extended include sensor error models experimenting attacks combining false data injection likely interesting path even though attacks sensors would rely additional equipment done manipulating vehicle malware also found injection attacks cause crashes vehicles behind attacker development future control algorithms suggest authors consider fallback mechanism similar ploeg controller however suggest eventually downgrades degraded cacc pure acc error estimated acceleration increase time jamming attack maintained long enough improve reliability using available inputs may feasible approach reduce impact attacks partially demonstrated consensus controller especially common controllers typically use leader preceding vehicle information may increase overall security make difficult successfully attacks achieve desired impact information show inconsistencies fallback mechanism could considered challenge ensuring entire platoon mutually decides performing fallback avoid heterogeneous controller environment attacks discussed work may detected relative ease using plausibility checks however main aim work show attack impact may maximized future work may look maximum tolerance attacks designed however currently proposed metrics bound behavior leader next step decide use information apart invalidating vehicle scms preventing damage ongoing attacks also goal using controller graceful degradation similar ploeg detection may trigger functionality combined effective revocation ensure temporarily degraded service best possible attack interesting side aspect detection algorithms compatible platooning corresponding behavior different may thus feasible path design specialized misbehavior detection mechanisms cacc scenario specifically acknowledgment authors would like thank johannes diebold thesis upon work expands henning kopp proof reading paper experiments work performed computational resource bwunicluster funded ministry science research arts universities state germany within framework program bwhpc work supported part stiftung ggmbh stuttgart part project autodetect security research programme eferences jia wang zhang shen survey platoonbased vehicular systems ieee communications surveys tutorials vol firstquarter rajamani vehicle dynamics control springer raya hubaux securing vehicular hoc networks journal computer security vol dadras gerdes sharma vehicular platooning adversarial environment proceedings acm symposium information computer communications security ser asia ccs new york usa acm debruhl weerakkody sinopoli tague commute driving crazy study misbehavior vehicular platoons proceedings acm conference security privacy wireless mobile networks ser wisec new york usa acm online available http amoozadeh raghuramu chuah ghosal zhang rowe levitt security vulnerabilities connected vehicle streams impact cooperative driving ieee communications magazine vol june dabaghchian zhang zeng string stability analysis cooperative adaptive cruise control jamming attacks ieee international symposium high assurance systems engineering hase jan segata joerer bloessl sommer dressler cigno plexe platooning extension veins ieee vehicular networking conference vnc paderborn germany ieee december ploeg lijster van wouw nijmeijer graceful degradation cacc performance subject unreliable wireless communication intl ieee conference intelligent transportation systems itsc oct bernardo salvi santini distributed consensus strategy platooning vehicles presence heterogeneous communication delays ieee transactions intelligent transportation systems vol feb van der heijden kargl almomani enhanced position verification vanets using subjective logic vehicular technology conference fall ieee gerdes winstead heaslip cps efficiencymotivated attack autonomous vehicular transportation proc annual computer security applications conference ser acsac new york usa acm petit shladover potential cyberattacks automated vehicles ieee transactions intelligent transportation systems vol apr feiri petit kargl certificate omission vanets intl workshop vehicular systems applications ser vanet usa acm douceur sybil attack systems ser lecture notes computer science druschel kaashoek rowstron vol springer berlin heidelberg mar joksch velocity change fatality risk crash rule thumb accident analysis prevention vol larburu sanchez rodriguez safe road trains environment human factors aspects dual mode transport systems world congress busan korea whyte weimerskirch kumar hehn security credential management system communications ieee vehicular networking conference dec
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sep classification artin groups finite case jingyin huang abstract let two artin groups show isomorphic assumption outer automorphism groups finite assume finite isomorphic subgroup finite index case give algorithm determine whether looking defining graphs contents introduction backgrounds summary results comments proof organization paper acknowledgement preliminaries notation conventions cat space cat cube complex coarse intersections convex subcomplexes artin groups stable subgraph coarse intersection standard subcomplexes flats standard flats transvection free raag standard flats general raag isomorphisms extension complexes reconstruction automorphism groups extension complexes special subgroups preservation extension complex coherent ordering coherent labeling geometry finite index raag subgroups constructing finite index raag subgroups rigidity raag subgroups generalized star extension references jingyin huang introduction backgrounds summary results given finite simplicial graph vertex set artin group raag defining graph denoted given following presentation joined edge called standard generating set section class raag enjoys balance simplicity complexity one hand raag many nice geometric combinatorial group theoretic properties see summary hand class inherits full complexity collection finite simplicial graphs even single raag could complicated subgroups see example recent years raag become important models understand unknown groups either virtually embedding unknown groups raag program outlined section see also references finding embedded copies raag unknown groups paper study asymptotic geometry raag classify particular class raag types previously quasiisometric classification raag done following two classes tree groups behrstock neumann shown two trees diameter quasiisometric higher dimensional analogs tree groups studied atomic groups bestvina kleiner sageev raag atomic defining graph connected contain valence one vertices cycles length separating closed stars shown two atomic raag isomorphic note atomic groups much rigid tree groups define dimension maximal contains subgroup coincides cohomological dimension atomic groups hence natural ask higher dimensional raag satisfy similar rigidity properties atomic raag starting point current paper since looking raag rigid ones small quasiisometry groups would reasonable candidates however even atomic case group huge see discussion flexibility section turn outer automorphism group guidance ask whether raag small outer automorphism groups also geometrically rigid appropriate sense actually small outer automorphism groups commensurability rigidity results come together several cases example higher rank lattices mapping class groups etc first result classification raag finite outer automorphism group theorem pick finite isomorphic classification artin groups theorem proved section see theorem detailed version theorem collection raag finite outer automorphism group reasonably large class recall correspondence finite simplicial graphs raag thus makes sense talk random raag considering model random graphs parameters model right range almost raag finite outer automorphism group class raag finite outer automorphism group strictly larger class atomic raag moreover plenty higher dimensional raag finite outer automorphism group whether finite easily read defined closed star vertex denoted full subgraph see section spanned vertices adjacent similarly defined full subgraph spanned vertices adjacent note definition slightly different usual one results generated following four types elements identify vertex set standard generating set given vertex sending fixing generators graph automorphisms vertices sending fixing generators induces group automorphism called transvection adjacent transvection otherwise transvection suppose disconnected one obtains group automorphism picking connected component sending vwv vertex generators fixed called partial conjugation elements type infinite order elements type finite order finite contain separating closed star exist distinct vertices theorem suppose finite following equivalent isomorphic subgroup finite index isomorphic denotes extension graph introduced kim koberda see definition extension graphs viewed curve graphs raag analog carries aspect rigidity namely mapping class group shown naturally acts curve graph associated still true raag restriction outer automorphism group example finite however general exists pair commensurable raag different extension graphs see example also exists pair raag isomorphic extension graphs see section jingyin huang motivated theorem look finite index raag subgroups subgroups also raag given raag necessarily finite outer automorphism group pick standard generating set let word metric respect subset three points must every finite subset naturally gives rise finite index raag subgroup fundamental domain left action example pick rectangle size corresponding subgroup form detailed construction general case given section called subgroup subgroup special standard generating set similar construction case coxeter groups alternating description terms canonical completion introduced let salvetti complex see section let universal cover pick identification subset gives rise convex subcomplex corresponding special subgroup fundamental group canonical completion respect local isometry next result says finite way obtain finite index raag subgroups theorem suppose finite let standard generating set finite index raag subgroups moreover correspondence finite subsets based identity finite index raag subgroups see theorem slight reformulation theorem need explain two terms based identity example take inside rectangle could fundamental domain action naturally require rectangle first quadrant contain identity would give unique choice similar things done raag two terms defined precisely section simple example correspondence finite index subgroups form intervals form corollary finite isomorphic special subgroup turns algorithm enumerate defining graphs special subgroups raag theorem finite obtained finitely many gse particular algorithm determine whether looking graphs gse generalized version star extension example see also lemma defined section question motivated theorem following classification artin groups question let raag finite let finite generated group say partial answer question prove following result theorem let question induced conjugate geometric action cat cube complex closely related comments proof proof theorem start several notations salvetti complex denoted universal cover denoted flats cover standard tori called standard flats see section precise definition terms let proof theorem follows scheme proof main theorem similar schemes also found three steps first show maps top dimensional flats top dimensional flats finite hausdorff distance however collection top dimensional flats large linked directly combinatorics raag second step show preserve standard flats finite hausdorff distance third step straighten actually maps standard flats standard flats exactly finite hausdorff distance conclusion follows automatically cases first step done show still preserves top dimensional flats finite hausdorff distance higher dimensional case assumption outer automorphism group needed step second step consists two parts first show preserves certain top dimensional maximal products finite hausdorff distance one wish pass standard flats intersecting top dimensional objects however higher dimensional case lower dimensional standard flat may intersection top dimensional objects even case intersection one may able read information directly defining graph quite different situation relies several new ingredients necessary condition preserve standard flats every elements implies could transvections condition also sufficient theorem suppose transvection free exists positive constant standard flat exists standard flat denotes hausdorff distance step introduce auxiliary simplicial complex serves link asymptotic geometry combinatorial structure precisely one hand viewed simplified tits boundary hand one read certain information wall space structure complex turns coincide extension graph introduced motivated viewpoint mapping class group jingyin huang denote tits boundary let union tits boundaries standard flats natural simplicial structure however contains redundant information seen similar situation link base point looks complicated essentially contain information redundancy resolved replacing spheres arise standard flats simplexes dimension gives rise well defined simplicial complex since standard flats exists standard flat see section properties theorem transvection free induces boundary map simplicial isomorphism next want consider converse reconstruct map boundary map following sense pick vertex let collection maximal standard flats containing theorem exists unique maximal standard flat one may wish map turns intersection intersection however general may empty contain one point hence map may turns also rule partial conjugations exactly point give rises map maps vertices standard flat vertices standard flat also finite define inverse map enough deduce theorem proof theorem assumed finite still recover fact simplicial isomorphism since theorem say standard flat find standard flat hence define however inverse exist general next step trying extend cubical map definition obvious obstructions though maps vertices standard geodesic vertices standard geodesic may preserve order vertices typical example given following picture one permute green level red level tree order vertices black line preserved classification artin groups remedy flip backwards namely sequence permutations levels resulting map restricted standard geodesic respects order extend cubical map argument relies understanding flexibility namely much room perform flips one formulation aspect following theorem finite aut isom denotes syllable metric see section theorem theorem rely cubical map particular vertex compact convex subcomplex obtain subset theorem organization paper section contains basic notations used paper background material cat cube complexes raag particular section collects several technical lemmas cat cube complex one skip section first reading come back needed section prove theorem section stability top dimensional maximal product subcomplexes section deals lower dimensional standard flats section prove theorem construct extension complex viewpoint section explain object related tits boundary flat space contact graph section describe reconstruct section section devoted proving theorem prove theorem corollary theorem section acknowledgement would like thank bruce kleiner robert young reading parts paper give valuable comments would thank jason behrstock ruth charney walter neumann suggestion interesting questions leads section paper would thank pallavi dani mark hagen thomas koberda kim kim related discussions also thank referee carefully reading paper providing helpful comments preliminaries notation conventions graphs paper simple flag complex graph denoted flag complex subcomplex combinatorial polyhedral complex full contains subcomplexes vertex set also call full subgraph use denote join two simplicial complexes denote join two graphs let simplicial complex graph viewing metric graph edge length obtain metric defined denote let subcomplex define orthogonal complement denoted set vertex define link denoted full subcomplex spanned define closed star denoted full subcomplex spanned suppose subcomplex denote closed star full jingyin huang subcomplex define similar way let arbitrary subset denote collection vertices inside use denote identity element group use denote identity map space let metric space open ball radius centred denoted given subsets open subset denoted diameter denoted diam hausdorff distance denoted also use following adapted notation coarse set theory introduced symbol meaning cat space cat cube complex standard reference cat spaces let cat space pick denote unique geodesic segment joining denote comparison angle alexandrov angle boundary denoted collection asymptotic classes geodesic rays angular metric defined lim unit speed geodesic rays emanating base point metric depend choice length metric associated angular metric denoted called tits metric tits boundary denoted cat space see chapter given two metric spaces denote cartesian product cat cat space image isometric embedding note flat convex pick convex subset also cat use denote nearest point projection moreover pick see proposition another convex set parallel constant functions case natural isomorphism convex hull define parallel set denoted union convex subsets parallel geodesic extension property generally section convex subset moreover admits canonical splitting also cat space classification artin groups turn cat cube complexes cube complexes paper assumed finite dimensional cube complex obtained gluing collection unit euclidean cubes isometrically along faces see precise definition cube complex natural piecewise euclidean metric metric complete geodesic since finite dimensional curved link vertex flag complex addition simply connected metric cat said cat cube complex put different metric considering metric graph edge lengths called metric use cat metric metric natural injection lemma paper mainly use cat metric unless otherwise specified also notions depend metric like geodesic convex subset convex hull etc understood automatically respect cat metric unless otherwise specified definition section cellular map cat cube complexes cubical restriction cubes factors first map natural projection onto face second map isometry geodesic segment geodesic ray geodesic isometric embedding respect cat metric combinatorial geodesic segment combinatorial geodesic ray combinatorial geodesic embedding image subcomplex let cat cube complex let subcomplex following equivalent see convex respect cat metric full subcomplex convex respect metric link full subcomplex every vertex collection convex subcomplexes cat cube complex enjoys following version helly property lemma let collection convex subcomplexes lemma let two cat cube complexes let convex subcomplex admits splitting convex subcomplex lemma clear general case follows special case come notion hyperplane cubical analog track introduced hyperplane cube complex subset connected cube either empty union minimal exists satisfying jingyin huang recall subset form one coordinate functions cat cube complex following true see hyperplane embedded either empty general cube complexes possible contains two convex subset induced cell structure also cat cube complex exactly two connected components called halfspaces closure halfspace called closed halfspace also convex respect cat metric let smallest subcomplex contains convex subcomplex natural isometry called carrier every edge exists unique hyperplane intersects midpoint case say hyperplane dual edge dual hyperplane lemma also true collection hyperplanes easy see edge path combinatorial geodesic segment exist two different edges dual hyperplane moreover two vertices distance exactly number hyperplanes separate pick edge let cat projection hyperplane dual exactly convex moreover parallel let nhe carrier hyperplane dual nhe closure alternatively describe nhe parallel set coarse intersections convex subcomplexes lemma lemma let cat cube complex dimension let convex subcomplexes put let empty convex maps isometrically onto maps isometrically onto cat convex hull isometric since taking cat convex hull subcomplex subcomplexes cubical isomorphism inverse given exists remark equation implies min moreover classification artin groups remark implies large enough use describe situation stands word intersect next lemma gives combinatorial description lemma let pick edge let hyperplane dual conversely hyperplane satisfies dual hyperplane edge moreover proof first part lemma follows lemma let pick let let carrier thus implies similarly hence argument thus lemma follows lemma remark lemma also applied cat rectangle complexes finite type whose cells form finite type means finitely many isometry types rectangle cells rectangle complex lemma let hyperplane separating exists convex set parallel proof let let convex hull want prove suffices show edge pick point let point since parallel sits inside cube parallel edge cube thus either parallel edge dual second case implies dual impossible done since exactly one point artin groups pick finite simplicial graph let raag generating set called standard generating set relators associated group presentation commutators standard generating set determines graph whose vertices elements two vertices adjacent corresponding group elements commute follows isomorphism type depend choice standard generating set particular isomorphic let standard generating set label vertices elements nice space called salvetti complex see curved cube complex usual presentation complex presentation complex contains copy attach obtain build inductively manner process stop finitely many steps closure torus kind called standard torus correspondence standard tori dimension complete subgraph jingyin huang vertices thus dim dim define dimension dimension denote universal cover cat cube complex previous labeling vertices induces labeling standard circles lifts labeling edges choose orientation standard circle would give directed labeling edges pick base point vertex correspondence words edge paths start full subgraph gives rise subgroup subgroup kind called subgroup left coset subgroup called coset omit generating set clear also embedding locally isometric let covering map connect component convex subcomplex isometric call components standard subcomplexes defining graph standard standard complex covers standard also call standard geodesic pick identification cayley graph hence identified vertices let base vertex corresponds identity cayley graph convex hull hgv standard subcomplex associated thus correspondence standard subcomplexes defining graph left cosets note every edge vertex shares label denote vertex subcomplex define edge full subgraph spanned called support particular standard subcomplex defining graph every finite simplicial graph admits canonical join decomposition maximal clique join factor allow join decomposition point irreducible join decomposition trivial decomposition induces product decomposition called rahm decomposition consistent canonical product decomposition cat cube complex discussed section turn asymptotic geometry raag artin group connected moreover infinity read see order classify raag suffices consider raag follows main results moreover lemma implies following lemma exists connected component point standard subcomplex defining graph unique connected component unique standard subcomplex defining graph shown linear divergence join one point implies join classification artin groups invariant moreover results together theorem implies rahm decomposition stable theorem given let let corresponding rahm decomposition exist constants factors projections thus order study classification raag suffices study raag irreducible rely finer invariant raag recall case gromov hyperbolic spaces map geodesics geodesics finite hausdorff distance hence induces boundary map analog fact raag established map finite hausdorff distance following higher dimensional generalization theorem theorem theorem dim dim constant flat unique flat artin group simplicial graph called extension graph introduced extension graphs viewed curve graphs raag definition definition vertex set consists words conjugate elements recall standard generating set two vertices adjacent corresponding words commute flag complex extension graph called extension complex since curve graph captures combinatorial pattern dehn twist flats intersect mapping class group plays important role rigidity mapping class group similarly see section extension graph captures combinatorial pattern coarse intersection certain collection flats raag invariant certain classes raag stable subgraph study behavior certain standard subcomplexes section coarse intersection standard subcomplexes flats lemma let finite simplicial graph let two standard subcomplexes also standard subcomplexes jingyin huang proof lemma clear assume pick vertex lemma exists vertex let unit speed geodesic find sequence cubes contains interior points let vbi recall vbi collection labels edges section let vki put denotes orthogonal complement see section let full subgraph spanned let standard subcomplex defining graph contains empty claim pick edge let hyperplane dual carrier since assume definition edge dual thus cobound isometrically embedded flat rectangle one side rectangle implies let side rectangle opposite define similarly define let edge path starting edge follows argument induction combinatorial length thus direction since convex subcomplex lemma suffices prove every vertex belongs induction argument need show edge lemma implies exists edge cobound isometrically embedded flat rectangle one side rectangle carrier hyperplane dual follows corollary let let hyperplane separating let edge dual particular pick vertex hyperplane separating edge dual proof let proof lemma let collection vertices edge satisfying suffices prove clear since hyperplanes separates intersects transversally one point see suffices show hyperplane intersects transversally let suppose let consider triangle recall since see proof lemma contradiction similarly remark recall standard coset left coset standard subgroup lemma implies pair standard cosets associated another standard coset captures coarse intersection pair moreover also define notion distance two standard cosets takes value classification artin groups lemma let convex subcomplex let support see section full subgraph spanned see section parallel set convex subcomplex canonically splits note require satisfy geodesic extension property proof pick vertex let let unique standard subcomplex passes defining graph recall denotes graph join let natural copy inside clear let convex subset parallel let isometry induced cat projection onto pick vertex let geodesic segment connecting define proof lemma note necessarily vertex let edge flat rectangle three sides thus contained carrier hyperplane dual note side opposite given edge find edge path first last edge induction combinatorial length argument show thus follows remark following generalization lemma general cat cube complexes let cat cube complex convex set regular space direction see chapter satisfies subcomplex respect canonical spherical complex structure exists isometric centred cone point euclidean cone regular convex subset convex admits splitting induced cubical structure cat lemma let let subcomplex isometric suppose dim dim exist top dimensional flats exist constant subcomplex isometric exists constant proof theorem exist top dimensional flats thus exists remark imply jingyin huang let exists lemma convex subcomplex together imply isometric finite intervals moreover diam must bounded terms thus follows let collection top dimensional flats contained parallel set lemma implies exists let top dimensional flat exists lemma imply thus exists follows running argument inverse tree product convex subcomplex splits product trees exists cubical isomorphism trees standard tree product tree product also standard subcomplex one check standard tree product defining graph join decomposition discrete thus one choose standard subcomplexes note every standard flat standard tree product every subcomplex isometric tree product lemma suppose dim let quasiisometry let top dimensional tree product tree factors exists standard tree product proof essentially follows theorem proof let vti collection labels edges case consist one point follows theorem contains least two points lemma geodesic exists subcomplex isometric since unique parallelism collection labels edges depend choice denoted define varies among geodesics claim see pick geodesic let exist top dimensional flat geodesic lines subcomplex since lemma assume subcomplex pick since orthogonal infinite hausdorff distance thus infinite hausdorff distance assumption isometric convex subcomplexes thus either parallel classification artin groups orthogonal former impossible since infinite hausdorff distance thus mutually orthogonal collection let discrete full subgraph dimension assumption let collection top dimensional flats let unique flat note arbitrary exists finite chain starts ends intersection adjacent elements chain contains top dimensional orthant thus collection also property contained standard subcomplex defining graph remains deal case exist suppose applying lemma reduce lower dimensional case lemma follows induction dimension corollary let let top dimensional maximal standard tree product properly contained another tree product exists standard tree product standard flats transvection free raag dealt top dimensional standard subcomplexes next goal study standard subcomplexes necessarily top dimensional particular interested whether preserve standard flats finite hausdorff distance answer turns related outer automorphism group one direction obvious namely maps standard flat standard flat finite hausdorff distance must transvection free contain transvections converse also true set several necessary tools prove converse section always finite simplicial graph definition subgraph stable full subgraph let standard subcomplex let finite simplicial graph exists standard subcomplex simplicity also say pair stable case standard subcomplex stable arises stable subgraph claim defining graph stable see pick graph pick standard subcomplex defining graph note isometry since map stability hausdorff close standard subcomplex hence true follows claim one obtain invariants identifying certain classes stable subgraphs jingyin huang immediate definition finite simplicial graphs stable stable stable however necessarily true stable stable sequel investigate several properties stable subgraph following lemma easy consequence lemma remark lemma suppose stable also stable following result follows lemma lemma stable every connected component contains one point also stable lemma suppose stable let vertex set let full subgraph spanned orthogonal complement also stable proof let standard subcomplex defining graph let standard subcomplex satisfying lemma implies vertex denote let exists standard subcomplex thus vertices follows lemma parallel thus moreover also standard subcomplex lemma considering inverse repeating previous argument know thus also stable lemma suppose stable pick vertex full subgraph spanned stable proof use denote full subgraph spanned let standard subcomplex let unique standard subcomplex pick vertex let edge suppose end point let standard subcomplex contains defining graph denote hyperplane dual since thus separates follows corollary particular depending dimension lemma follows since stable next result direct consequence corollary lemma stable exists stable graph join discrete dim lemma let finite simplicial graph exist vertices every stable subgraph contains stable vertex classification artin groups proof let minimal stable subgraph properly contain stable subgraph suffices show point argue contradiction assume contains one point first claim discrete suppose contrary true pick vertices pick vertex lemma also stable note contradicts minimality claim must clique since discrete lemma find stable subgraph discrete full subgraphs suppose contains one point let join remaining join factors theorem implies stable contradicting minimality therefore clique pick distinct vertices assumption exists vertex since clique let full subgraph spanned stable lemma moreover since yields contradiction lemma let lemma let stable subgraph vertex exists stable vertex proof denote combinatorial distance respectively since full subgraph vertices isolated use argument second paragraph proof lemma get rid vertices except implies stable vertex isolated assume connected lemma lemma exists stable vertex done otherwise let geodesic connecting might geodesic let consecutive vertices since stable lemma also stable note lemma implies stable lemma also stable note since contains lemma implies stable vertex easy see lemma follows induction lemma let lemma every vertex stable proof let intersection stable subgraphs contain lemma minimal stable subgraph contains suffices prove argue contradiction denote vertices minimality implies use lemma get rid keep thus words exist otherwise would jingyin huang hand lemma implies exists stable vertex stable lemma stable lemma note minimality yields contradiction lemma let finite simplicial graph pick stable subgraphs let full subgraph spanned full subgraph spanned stable simplify notation following proof denote definition also assume without loss generality proof let suppose standard subcomplexes put proof lemma implies exist constant projections let let standard subcomplexes suffices prove exist standard subcomplex constant let standard subcomplex thus take remark general full subgraph spanned necessarily stable even stable see remark next theorem follows lemma lemma theorem given finite simplicial graph following equivalent transvection free exists positive constant standard flat exists standard flat standard flats general raag point following natural questions theorem true every standard flat comes standard flat related question could condition theorem invariant artin groups say stable subgraphs drop condition theorem first give negative answer question example prove theorem answers question section particular proof theorem depend subsection however need theorem lemma section classification artin groups theorem let arbitrary finite simplicial graph clique stable exist vertices words clique stable corresponding subgroup invariant transvections example let graph left let one right easy see transvection free contains nontrivial transvection dead end vertex claim commensurable particular let pentagon left side let salvetti complex suppose two boundary circles annulus identified two standard circles different copies define homomorphism sending element generators identity element let cover respect ker define homomorphism sending element generators identity element let cover respect ker made two copies two annuli boundaries annuli identified two cover see picture homotopy equivalent salvetti complex see let circle covers two times let wedge two circles covers wedge copy inside let segment end points mapped base point covering map collapse inside copy collapse one annuli circle killing interval factor denote resulting space homotopy equivalent annulus becomes torus hard see salvetti complex defining graph standard geodesic comes vertex hausdorff close image standard geodesic since stable vertex every vertex stable jingyin huang generalization example suppose finite simplicial graph exist vertices separated intersection links define homomorphism sending element killing generators ker also artin group argument find defining graph let components suppose define full subgraphs moreover let graph obtained gluing two copies along let join one point defining graph ker obtained gluing along note taking advantage separating closed stars constructing counterexample separating closed stars allowed positive answer question see section rest subsection prove theorem arbitrary finite simplicial graph rest subsection theorem actually consequence following general result lemma pick vertex let intersection stable subgraphs contain define full subgraph spanned words minimal standard subgroup containing property invariant transvection show deduce theorem lemma proof theorem part proved contradiction choose transvection preserve subgroup converse let vertex set let minimal stable subgraph contains assumption lemma thus full subgraph spanned stable lemma means stable remains prove lemma first set several auxiliary lemmas lemma let vertex isolated least one following true classification artin groups contained stable discrete subgraph one vertex contained stable clique subgraph stable discrete subgraph one vertex whose vertex set stable clique subgraph whose vertex set proof since isolated assume connected lemma lemma find stable subgraph discrete full subgraphs dim third paragraph proof lemma know either true suppose let full subgraph spanned stable lemma proof lemma implies every stable subgraph contains either stable discrete subgraph stable clique subgraph depend assumption thus either true suppose pick vertex let geodesic connecting suppose consecutive vertices let full subgraph spanned let full subgraph spanned vertex set stable lemma stable lemma note vertex vertex thus however induct reduce case interesting see large diameter lot nontrivial stable subgraphs record following lemma easy consequence theorem lemma suppose maximal clique join factor stable stable ready prove lemma proof lemma lemma minimal stable subgraph contains exists vertex sending fixing vertices would induce group automorphism gives rise existence would contradict stability thus let vertex set remains prove suppose let minimality implies use lemma get rid keep summary particular isolated apply lemma recall subgraph stable stable case lemma true get contradiction since isolated case true sits inside clique contradictory case true let corresponding stable discrete subgraph let let suppose jingyin huang full subgraph spanned stable lemma hence let join decomposition induced rahm decomposition sit inside clique factor clique factor join factor stable theorem inside one join factors contradict minimality clique factor exists sits inside clique factor theorem clique factor stable contradiction clique factor exists sits outside clique factor reduces next case case true let corresponding stable clique subgraph also assume without loss generality contained stable clique let suppose full subgraph spanned let corresponds euclidean rahm factor note discussion case equation implies still true take orthogonal complement closed star particular isolated moreover dim dim dim dim discrete contradictory isolated dim induction assume lemma true lower dimensional graphs exists stable lemma stable hence contradicts minimality remark nature ask whether theorem still true require clique turns counterexamples let graph disjoint union easy check exist note separates get partial conjugation sends implies stable interesting example nature following let graph left side graph right side discussion section let let standard subcomplex defining graph pentagon suppose hausdorff close standard subcomplex must connected proper subgraph hence tree impossible results classification artin groups isomorphisms extension complexes extension complexes standard flats let quasiisometry usually induce boundary map see however theorem implies still control subset tits boundaries transvection free subsection reorganize piece information terms extension complexes recall identify vertex set standard generating set also label standard circles salvetti complex elements choosing orientation standard circle obtain directed labeling edges denote extension complex give alternative definition natural purposes vertices correspondence parallel classes standard geodesics two standard geodesics parallel class parallel two distinct vertices connected edge find standard geodesic parallel class associated span standard next observation follows lemma lemma observation joined edge exist parallel class associated defined flag complex lemma isomorphic extension complex proof suffices show isomorphic extension graph pick vertex let standard geodesic parallel class associated identify way orientation induced directed labeling recall deck transformations let element easy see conjugate element thus gives rise vertex definition note depend choice parallel class map vertex set vertex set moreover adjacent commute define inverse map pick gsg standard geodesics stabilized parallel class let vertex associated parallel class map vertex vertex show map extends let centralizer theorem commute jingyin huang exists gsi thus adjacent since every edge standard geodesics parallel class label labeling edges induces labeling vertices moreover since cubical isomorphisms obtain induced action simplicial isomorphisms moreover unique map vertices vertices extends simplicial map pick arbitrary vertex one obtain simplicial embedding flag complex considering collection standard geodesics passing denote image note vertex identity map pick vertex set pick standard geodesic parallel class associated since pli plj lemma pli corollary lemma exist standard geodesics satisfying parallel convex hull standard denoted pli pfk thus correspondence parallel classes standard particular maximal simplexes namely simplexes properly contained larger simplexes correspondence maximal standard flats standard flat denote simplex associated parallel class containing observation let two simplexes let standard flat set define reduced tits boundary denoted subset union tits boundaries standard flats standard flat triangulate spherical simplexes tits boundaries orthant subcomplexes pick another standard flat subcomplex lemma remark thus endow structure spherical complex look relation standard flat associate induces surjective simplicial map defined induction dimension note inverse image simplex sphere one construct follows start collection correspondence vertices form join copies corresponding vertices span simplex words obtained applying spherical complex construction sense definition classification artin groups let standard subcomplex define union tits boundaries standard flats note descends subcomplex denoted lemma let two standard subcomplexes put proof remark hence study extension complexes behave lemma pick transvection free induces simplicial isomorphism assumed transvection free still simplicial embedding proof proof case transvection free case similar theorem every vertex stable thus sends parallel class standard geodesics another parallel class standard geodesics finite hausdorff distance induces map map bijection considering inverse moreover observation implies two vertices adjacent images adjacent extend graph isomorphism since flag complexes extend isomorphism whole complex extension complexes relatives discuss relation several objects literature material subsection used later endow structure complex groups gives alternative definition specifically vertex identified exists integer also view one generators hence vertex set identified belongs subgroup generated one compare similar construction coxeter group another important object associated artin group called modified deligne complex flat space definition let poset left cosets standard abelian subgroups include trivial subgroup partial order induced inclusion sets modified deligne complex defined geometric realization derived poset recall elements derived poset poset totally ordered finite chains viewed abstract simplex extension complex viewed coarse version modified deligne complex let two subsets metric space say coarsely equivalent coarsely contained let poset whose elements coarsely equivalent classes left cosets jingyin huang standard abelian subgroups partial order induced coarse inclusion sets observation poset abstract simplicial complex isomorphic roughly speaking captures combinatorial pattern standard flats intersect coarsely intersect thus contains information however certain cases possible recover information enable prove results raag define poset arbitrary artin group considering collection coarse equivalent classes spherical subgroups artin group coarse inclusion geometric realization derived poset would natural candidate extension complex artin group interesting ask much results generalized context also link structure hyperplanes recall every cat cube complex crossing graph denoted graph whose vertices correspondence hyperplanes two vertices adjacent corresponding hyperplanes intersect contact graph introduced denoted vertex set two vertices adjacent carriers corresponding hyperplanes intersect natural surjective simplicial map defined follows pick vertex let corresponding hyperplane since standard geodesics intersect one point parallel class define vertex associated parallel class see lemma clear adjacent vertices adjacent extends simplicial map pick vertex collection hyperplanes dual standard geodesic theorem connected moreover tree viewpoint captures geometric information standard flats combinatorial information hyperplanes reconstruction show boundary map lemma induces well defined map lemma let two maximal standard flats let corresponding maximal simplexes separated hyperplane exist vertices different connected component proof let edge dual let standard geodesic contains set lemma parallel set isometric thus every standard geodesic parallel must intersection since contain standard geodesic parallel means moreover since maximal simplex similarly thus find vertices claim vertices looking classification artin groups path connecting assume consist sequence edges pick maximal simplex contains let maximal standard flat hence set since contains vertex since different sides exists separates let observation however lemma exists convex subset parallel thus follows observation contradicts denote cayley graph respect standard generating set pick identification thus identified vertex set lemma let finite simplicial separating closed star contained union two closed stars simplicial isomorphism induces unique map maximal standard flat vertices mapped vertices lying maximal standard flat proof pick vertex let collection maximal standard flats containing define let maximal standard flat let recall claim lemma follows see deduce condition hence follows together imply exactly one point define sending point one readily verifies required properties remains prove suppose true lemma exist thus separated hyperplane follows lemma exist vertices different connected components let disconnected since separated would contain separating closed star yields contradiction thus true case suppose pick standard geodesic let collection hyperplanes separates parallel set note pick edge dual let unique vertex label let jingyin huang unique vertex label claim every let unique standard geodesic pick observation implies standard geodesic separates otherwise lemma implies contradiction follows corollary adjacent thus therefore pick first show suppose vertex since label follows edge contains belongs parallel set contradicting fact therefore pick edge path shortest combinatorial length travels let consecutive edges let hyperplane dual separates otherwise would shortest edge path hence separates imply vfj map vfj label edge follows contained parallel set plu hence intersection plu contains vertex since label find standard geodesic plu parallel passes contained therefore follows condition lemma let let implies thus following sequence reduced homology recall disconnected deduce nontrivial thus nontrivial implies either would separate thus induct deduce exists separates yields contradictory condition lemma counterexamples assume lemma example let discrete graphs made two points discrete sets hard construct permutation discrete set satisfy conclusion lemma back proof lemma step using sequence fail since need order use reduced version sequence corollary suppose satisfy assumption lemma isomorphic isomorphic simplicial complexes proof direction follows fact isomorphic isomorphic see remains prove classification artin groups direction pick isomorphism let map lemma pick vertex let define first paragraph proof lemma implies induces graph embedding repeating previous discussion obtain another graph embedding since finite simplicial graphs isomorphic hence lemma let raag finite satisfies assumption lemma proof clear satisfy condition lemma since nontrivial partial conjugation allowed contained closed star point true distinct vertices since orthogonal complement satisfies exists pick edge implies hence infinite yields contradiction lemma lemma corollary following result particular establishes theorem introduction theorem let two finite simplicial graphs finite isomorphic moreover exist bijection constant standard flat exists standard flat induces bijection unique proof suffices look case case every vertex intersection maximal cliques contain otherwise exist vertex follows every standard geodesic intersection finitely many maximal standard flats every standard flat let map lemma apply lemma obtain required properties note vertex intersection maximal standard flats contain thus unique automorphism groups extension complexes suppose finite theorem element simplicial automorphism group aut induces bijection however bijection extend isomorphism general start looking following example first pointed section slightly different form example let standard geodesic let cat projection identify vertex set let vertex set projection induces map jingyin huang recall edge oriented labeled acts transformations preserve labels orientations unique element translates one unit towards positive direction want define bijection basically flips precisely one check following respect word metric maps vertices standard flat vertices another standard flat thus induces element aut example implies general elements aut respect order along standard geodesics another metric forgets ordering following define syllable length word minimal written product elements form viki standard generator integer alternative definition following let collection hyperplanes separating identity element recall identified pick standard geodesic dual syllable length number elements syllable length induces left invariant metric denoted note map example isometry respect denote word metric respect standard generators corollary let graph finite denote simplicial automorphism group aut aut isom proof group homomorphism aut erm lemma erm permutation group elements take aut lemma satisfy conclusion lemma since every standard geodesic intersection finitely many maximal standard flats points standard geodesic mapped points standard geodesic implies triangle inequality similarly thus isom homomorphism aut isom pick isom let claim find vertices flat rectangle note consider formed angles four vertices classification artin groups bigger equal follows cat geometry angles exactly actually bounds flat rectangle thus one direction proved direction similar need another observation follows three points satisfies angle vertex triangle could thus inside standard geodesic follows observation points standard geodesic mapped points standard geodesic define follows vertex let standard geodesic suppose standard geodesic denotes vertex set suppose define implies depend choice adjacent vertices adjacent thus simplicial map note also induces simplicial map similar way aut define one readily verify isom aut group homomorphism thus corollary follows remark drop assumption corollary still monomorphism isom aut moreover isom maps vertices standard flat vertices standard flat dimension homomorphism surjective finite remark finite simplicial graphs isometric metric spaces direction follows direction let isometry pick let collection standard geodesics passing pick induces graph embedding considering obtain another graph embedding hence isomorphic corollary finite following commutative diagram injective homomorphisms isom isom group proof homomorphism obvious given lemma corollary clear group homomorphism note injective injective pick corollary know implies image every standard flat uniformly hausdorff close thus bounded distance identity map jingyin huang special subgroups let raag finite outer automorphism group section characterize raag preservation extension complex lemma let finite simplicial graph pick vertex let minimal stable subgraph containing denote see section definition links either following true clique case stable subgraph stable subgraphs moreover disconnected recall use denote orthogonal complement subgraph see section assume proof clique lemma also deduce since lemma vertex moreover thus full subgraph spanned vertices stable lemma let full subgraph spanned vertices let full subgraph spanned vertices lemma note disconnected isolated point may empty let vertex set let full subgraph spanned stable lemma pick vertex vertex lemma thus hand since contain clique factor stable know stable theorem remark proof may empty contain clique join factor thus maximal clique join factor next result answers question end example theorem suppose finite let quasiisometry induces simplicial isomorphism particular transvection free following proof identify flag complex also recall projections proof lemma simplicial embedding note full subcomplex see pick simplex vertices vertex comes stable standard geodesic line thus exists stable standard flat lemma considering inverse know hausdorff close stable standard flat thus classification artin groups pick vertex let proof lemma claim suppose true exist hyperplane separates let standard geodesic intersects transversely let discussion lemma find vertices separated exists prove lemma assume let let minimal stable subgraph contains apply lemma case true let standard flat since stable particular prove lemma case true let let take standard subcomplexes defining graphs satisfy set let orthogonal complement standard subcomplex follows construction since stable exist stable standard subcomplexes moreover applying theorem exists standard subcomplex thus also disconnected let stage may know see pick simplex suppose stable standard flat hence let proof lemma disconnected thus disconnected recall assuming thus hence let let separates disconnected pick vertices different connected components since full subcomplex spanned since let connected components one contained would separate contradiction suppose note must exist otherwise would separate moreover assume without loss generality implies contradiction lemma thus case impossible true let collection maximal standard flats let unique maximal standard flat jingyin huang let arbitrary hyperplane otherwise would stay one side hyperplane since connected set contradicts pick standard geodesic let hyperplane dual exists follows implies surjective vertices however full subcomplex surjective coherent ordering coherent labeling throughout section assume finite definition induces group homomorphism hand since acts isometries identify subgroup precisely embed isom embed isom corollary subsection understand following question exist recall picked identification circle salvetti complex labeled element standard generating set moreover chosen orientation circle pulling back labeling orientation edges universal cover obtain directed labeling edges moreover labeling orientation edges compatible parallelism edges also induces associated labeling vertices let collection standard geodesics let vertex set coherent ordering obtained assigning collection bijections parallel translation map induced parallelism map pulls back total order denote two coherent ordering equivalent denoted collections bijections agree translation recall invariant orientation edges compatible parallelism edges induces unique coherent ordering equivalence relation defined moreover element pull back also coherent ordering moreover recall vertex simplicial embedding considering standard geodesics passing coherent labeling simplicial map simplicial isomorphism every vertex projection gives rise coherent labeling recall acts simplicial automorphisms labeling vertices thus element pull back also coherent labeling classification artin groups following alternative characterization elements isom lemma correspondence associates element isom triple consisting point coherent ordering equivalence relation defined coherent labeling proof pick isom let monomorphism remark coherent labeling pick standard geodesic parallel set admits splitting since maps vertices standard flat bijectively vertices standard flat exists standard geodesic moreover respects product structure thus coherent ordering set correspondence one direction denotes identity element conversely given point coherent ordering coherent labeling construct map follows set pick word representing let point represented word let define inductively follows set suppose already defined denote standard geodesic containing let vertex let standard line contains labeled denote vertex set order suppose unique order preserving bijection define claim word representing hence map see recall one obtain performing following two basic moves commute clear second move let points represented respectively define since coherent labeling moreover standard geodesic containing parallel standard geodesic containing since coherent ordering thus parallel similarly parallel thus define another map serves inverse set pick word let point represented define inductively follows put suppose already defined since coherent labeling exists unique standard geodesic containing edge share label let unique standard geodesic containing let jingyin huang unique order preserving bijection put similar argument hard deduce following properties construction vertices left translation addition follows isom moreover monomorphism remark thus established required correspondence pick finite simplicial graphs finite exists simplicial isomorphism lemma induces map every left translation gives rise simplicial isomorphism let gives rise map isom corollary moreover lemma acts define homomorphism isom sending injective since step defining injective lemma setting exists element isom conjugates image finite index subgroup identify subgroup isom via left action proof pick reference point let denote points let since distinct subcomplexes finite set let coherent labeling coherent ordering induced labeling obtain coherent labeling coherent ordering similar fashion invariant goal find coherent labeling coherent ordering let canonical embedding let simplicial map pick arbitrary let canonical embedding need show simplicial isomorphism let let thus classification artin groups simplicial isomorphism lemma follows coherent labeling moreover third equality follows required coherent labeling simplify notation write ordering define follows let two distinct points standard geodesic line set exists unique set follows construction standard geodesic line thus verify coherent pick parallel standard geodesics pick distinct vertices let corresponding vertices via parallelism assume suffices prove case assume recall realized intersection finitely many maximal standard flats lemma exists standard geodesic line moreover respects product structures thus opposite sides flat rectangle follows since coherent case assume case assume without loss generality since points stay standard geodesic let standard geodesic passing take standard geodesics respectively denote since restricted respects product structure let left translation since translation along fixes every point hence fixes every point let parallel see note hence fixed put hence opposite sides flat rectangle moreover since coherent restricted implies case assume without loss generality equal follows since respects product structure restricted thus definition jingyin huang lemma exists isom monomorphism remark thus similarly note thus lemma acts left translations via induces monomorphism moreover fact finite action finite quotient thus realize finite index subgroup next result basically says suitable conditions exists quasiisometry exists nice however insist bounded distance away compared theorem theorem let finite simplicial graphs finite exists cubical map definition map onto maps standard flat onto standard flat dimension map maps combinatorial geodesics combinatorial geodesics map proof let theorem induces simplicial isomorphism lemma induces map let map lemma let use notation proof lemma claim maximal standard flat exists unique standard flat see let maximal standard flat let follows lemma recall acts stabilizer stab fixes hence fixes stab stab since stab acts transitively implies also follows stab stab thus stab stab note claim also true standard geodesic satisfies assumption claim moreover surjective since surjective pick standard geodesic identify way claim construction imply form integers particular extended simplicial map cayley graph pick combinatorial geodesic connecting vertices claim also geodesic could point let vertices maximal classification artin groups contained standard geodesic denote corresponding standard geodesic let standard geodesic possibly degenerate segment since geodesic none two geodesics parallel note induced simplicial isomorphism thus property true collection geodesics follows hyperplane could intersect one point hence combinatorial geodesic let recall denotes word metric corresponding group thus pick let depend follows whenever cut pieces length since combinatorial geodesic note naturally extends cubical map satisfies required properties theorem finite simplicial graphs finite following equivalent isomorphic simplicial complexes isomorphic subgroup finite index proof follows theorem follows lemma trivial establishes theorem introduction geometry finite index raag subgroups throughout section assume since main results section theorem theorem trivial constructing finite index raag subgroups artin subgroup subgroup also artin group section introduce process obtain finite index raag subgroups arbitrary raag lemma let cat cube complex let geodesic let consecutive hyperplanes dual let cat projection every edge vertex edge connected subcomplex vertex moreover stays vertex stays jingyin huang every interval convex set particular vertex convex subcomplex convex subcomplex proof follow fact every hyperplane carrier follows see suffices show edge dual let nhi carrier lemma nhi moreover nhi interior must otherwise convexity would imply lemma let standard geodesic map recall collection vertices connected component proof let cat projection let standard geodesic vertex lemma corollary moreover claim standard geodesic parallel suffices prove case unique hyperplane separating note yields pinched two hyperplanes dual claim follows lemma thus induces map connected edge exist standard geodesics parallel thus pick standard generating set let cayley graph identify subset attach higher dimensional cubes obtain cat cube complex basically universal cover salvetti complex would like think fixed set objects formed adding edges cubes particular way determined write explicitly choose equivariant orientation edges defined vertex set standard flat geodesic define vertices correspond coarse equivalence classes define isometric embedding depends orientation edges pick standard geodesic let cat projection identify orientation preserving way identity element induces coordinate function change standard geodesic parallel identical lemma thus every vertex coordinate function coordinate functions induce map embedding since every two points separated hyperplane since finitely many hyperplanes separating naturally extends map maps combinatorial geodesics geodesics argument theorem thus isometric embedding respect metric classification artin groups say convex subcomplex point coordinates notion depends orientation edges let collection compact convex subcomplexes contain identity find maximal collection standard geodesics let label edges let put gini let subgroup generated follows convexity standard geodesic parallel thus depend choice lemma finite index subgroup proof prove showing let syllable metric defined section pick word assume exists let suppose standard geodesic containing exists parallel note convex set parallel set hence respects natural splitting moreover left action translates factor units fixes factor thus exists vid implies vid let full subgraph spanned points natural homomorphism lemma homomorphism actually isomorphism hence finite index raag subgroup follow strategy following version lemma artin groups used theorem theorem let let set suppose following hold vertex exists subset union properly contained nonzero vertices joined edge vik nonzero vertices joined edge vik exists vertex set nonzero vik faithful proof lemma apply theorem identify orientation preserving way corresponds define clear identity element theorem true gini translates units also true jingyin huang connected edge stabilizes every hyperplane dual thus vik true point lemma yield thus boundary let similarly let carrier implies empty intersection hyperplane dual follows lemma vertex two hyperplanes pinch yield hence lemma similarly note vik vik vik kni vik discussion subsection yields map finite index raag subgroups images called subgroups subgroup special standard generating set rigidity raag subgroups subsection assume finite index raag subgroup finite show condition must arise process described previous subsection prove steps first produce convex subcomplex modify convex subcomplex element thus defined map finite index raag subgroups elements last step show map defined step inverse map defined section also near end subsection leave several relatively long remarks discuss relevant material literature reader skip remarks first reading recall finite transvection free theorem two standard generating sets differ sequence conjugations partial conjugations given two standard generating sets canonical way identify every hausdorff close thus write omit generating set lemma let discussion lemma let standard geodesics proof pick standard geodesics lemma recall intersection maximal standard flats therefore definition suffices show vertex let vertex let lemma approximate combinatorial geodesic hyperplane could intersect let vertices maximal classification artin groups contained standard geodesic denote corresponding standard geodesic let let standard geodesic thus point corollary step produce convex subcomplex left action induces choosing standard generating set left action use denote action respectively pick theorem lemma induces surjective maps note depends choice generating set flexibility comes automorphism groups key step choose nice standard generating set behaves like theorem lemma choosing possibly different standard generating set assume map satisfies denotes identity element corresponding group proof assume claim change generating set resulting satisfy requirement construction suffices show maximal exists maximal let assume would satisfy required condition prove general case similar way pick standard geodesic want flip order points way true choose order preserving identification let vertex let stab stabilizer action second paragraph proof theorem depend choice stab acts way acts recall action induced left action write parallel thus since depends stab acts depend standard generating set however choice descends let stab generated reasoning lemma assume let associated integer follows set consider case let standard geodesic contains powers let map lemma defined unique integer recall define sending extends automorphism also standard generating set indeed stay connected component jingyin huang lemma hence follows realized composition partial conjugations lemma replace definition denote new map satisfies standard geodesic recall use denote action proof suffices show satisfies rest follows equivariance show need prove let pick exists lemma hence let standard geodesic exists passing note look new map note still true moreover lemma imply thus lemma follows next lemma basically says change basis process affect geodesics essential way lemma let standard geodesic satisfies condition pick two different vertices proof let standard geodesic containing let resp resp geodesic resp let exist elements note recall generator stab first inequality follows suffices show exist follows lemma define immediate note follows definition classification artin groups similarly prove change respect conjugation lemma still true arbitrary standard geodesic lemma lemma apply procedure finitely many times find appropriate standard generating set corresponding map satisfies restricted standard geodesic proof theorem extend cubical map combinatorial geodesics mapped combinatorial geodesics thus combinatorially convex subcomplex subcomplex also compact since contains finitely many vertices recall combinatorial convexity convexity cat metric subcomplexes cat cube complexes constructed compact convex subcomplex given finite index raag subgroup step show assumed element denote union standard geodesics intersection vertex set convex subcomplex return step assume step let collection standard geodesics passing let map defined section since different lemma lemma apply procedure step find standard generating set step let pick vertex exist process terminate let standard geodesic parallel step replaced automatically true without modification respect product structure pli parallel moreover modify true deal standard geodesics passing points similar way step let vertex repeat procedure step define since finite number change adjusting procedure must terminate finitely many steps since remains connected step procedure terminates must already dealt point standard geodesic passing point construction resulting satisfies standard geodesic intersects thus note sets actually depend map step depend map thus subset produced depends subgroup map finite index raag subgroups jingyin huang step show inverse map defined section first prove let let corresponding standard generating set let corresponding map find maximal collection standard geodesics let let label edges suppose cat projection suffices prove following lemma lemma gini proof pick let standard geodesic containing exists unique see let standard geodesic parallel exists sends thus defined beginning step note nontrivial intersection choose geodesic parallel standard geodesic parallel parallel stab stab follows stabilizes parallel set pci acts translation along note pci gini claim follows remains show follows following result lemma let arbitrary finite simplicial graph pick standard generating set let let corresponding generating set suppose identity map induces simplicial isomorphism induces retraction sends every extends surjective cubical map particular vertex set strict fundamental domain left action proof suffices prove case admit nontrivial join decomposition point construction know intersects hausdorff close contains identity moreover maximal corresponding unique since equivariance implies every hausdorff close since images parallel hausdorff closed induces injective since surjective pick let collection maximal containing let unique maximal note assumption either empty one point note equivariance implies every point defined clear follows strict classification artin groups fundamental domain left action map maps note implies thus retraction similarly using equivariance deduce sends every intersects passing identity element thus sends every equivariance easy see extends cubical map remark generalize results lemma infinite convex subcomplexes convex subcomplex admissible standard geodesic cat projection either finite interval whole ray allowed let maximal collection standard geodesics parallel finite interval let element translates along translation length length let subgroup generated admissible prove moreover finite subset subgroup generated artin group isometric embedding respect word metric define view vertex show strict fundamental domain action suffices show nontrivial assume artin group let canonical form see section belongs abelian standard subgroup let exists commute associate generator subset proof lemma claim exists follows prove induction assume commute moreover define map sending identity prove maps possibly lower dimensional thus respect word metric let inclusion equivariance left translation particular contains identity retraction follows finite embedding note related construction case coxeter groups discussed taking larger larger convex compact subcomplexes know residually finite moreover pick stab definition stab normalize obtained direct proof fact every subgroup finite generated artin group separable theorem using discussion together outline section following result follows readily discussion jingyin huang theorem let raag finite pick standard generating set correspondence convex compact subcomplexes contain identity finite index raag subgroups particular subgroups generated conjugates powers elements particular theorem introduction follows theorem remark drop finite automorphism group assumption theorem exist raag finite index raag subgroup isomorphic special subgroup see let artin group transvection free lemma theorem imply special subgroup admit transvection outer automorphism group let graphs example artin subgroup transvections thus isometric special subgroup remark pick finite theorem used show certain subgroup raag example let subset standard generating set define homomorphism sending element killing generators ker raag one compare example example remark shown theorem embeds full subcomplex exists monomorphism result recovered previous discussion follows let arbitrary finite simplicial graph let standard generating set vertex let conjugate element every standard geodesic suppose compact full subcomplex denote vertex set let standard geodesic identify way identified cat projection identity element define define pair integers follows let minimal interval recall identified pick arbitrary set define construction satisfying using argument section show subgroup generated raag defining graph point natural ask following question question let standard generating set let finite collection elements form suppose subgroup generated artin group generalized star extension goal subsection find algorithm determine whether given finite classification artin groups convex subcomplex denote full subcomplex spanned collection standard geodesics describe process construct graph isomorphic special subgroup let let one point construct pair inductively compact cat cube complex cubical embedding convex finite simplicial graph simplicial isomorphism note assumptions true associate edge vertex denoted follows let standard geodesic contains define vertex associated full subcomplex defined define pick vertex let collection vertices exactly vertices standard geodesic let convex hull edge since natural product decomposition induces product decomposition note possible exists edge isomorphic hyperplane dual edge projection interval factor edge let let vertex define simplicial complex obtained gluing along see section notation define cat cube complex obtained gluing along one readily verifies one extend cubical embedding convex also induces isomorphism extension construction isomorphic special subgroup moreover associated convex subcomplex special subgroup also note induction process actually depend knowing thus also provides way construct convex subcomplexes hand process obtaining called generalized star extension gse note following equivalent standard geodesic natural projection defined gse nontrivial clique stage exists vertex gse nontrivial lemma suppose isomorphic special subgroup construct using finitely many gse jingyin huang proof let objects defined section suppose isomorphic define sequence convex subcomplexes induction let identity element suppose already defined induction terminates pick edge vertex let convex hull let resulting collection convex subcomplexes alternative way describing following hyperplane dual carrier convexity thus disjoint hence copy inside denoted one readily verifies one obtain gse construction gives rise algorithm detect whether isomorphic special subgroup vertices obtained nontrivial gse start enumerate possible nontrivial gse compare resulting graph theorem theorem following result theorem finite obtained finitely many gse particular algorithm determine whether note gse gives rise pair one care associated convex subcomplex simpler description gse finite suppose already obtained together finite collection full subcomplexes covering isomorphic pick trivial cover construct pick vertex let let suppose collection connected components suppose defined gluing along lemma suppose finite simplified process consistent gse proof assume inductively cat cube complex two induction assumptions gse satisfied moreover coincides vertex let let integer suffices show correspondence exists unique pick adjacent vertices let label edge suppose since finite orthogonal complement satisfies vertex lifts point contains vertex since separating closed stars connected thus connected follows connected moreover lemma implies classification artin groups different components exists unique references aaron abrams noel brady pallavi dani moon duchin robert young pushing fillings artin groups journal london mathematical society mladen bestvina noel brady morse theory finiteness properties groups inventiones mathematicae jason behrstock ruth charney divergence quasimorphisms artin groups mathematische annalen martin bridson haefliger metric spaces curvature volume grundlehren der mathematischen wissenschaften fundamental principles mathematical sciences berlin jason behrstock tadeusz januszkiewicz walter neumann classification high dimensional artin groups groups geom hyungryul baik kim thomas koberda artin subgroups interval diffeomorphism group arxiv preprint jason behrstock bruce kleiner yair minsky lee mosher geometry rigidity mapping class groups geom mladen bestvina bruce kleiner michah sageev asymptotic geometry artin groups geom mladen bestvina bruce kleiner michah sageev quasiflats cat complexes arxiv preprint noel brady john meier connectivity infinity right angled artin groups transactions american mathematical society jason behrstock walter neumann classification graph manifold groups duke mathematical journal noel brady tim riley hamish short geometry word problem finitely generated groups springer science business media ruth charney michael davis finite artin groups prospects topology princeton volume ann math pages princeton univ press princeton ruth charney michael farber random groups arising graph products algebr geom topol ruth charney introduction artin groups geometriae dedicata christopher croke bruce kleiner spaces nonpositive curvature ideal boundaries topology matt clay christopher leininger johanna mangahas geometry right angled artin subgroups mapping class groups arxiv preprint caprace nicolas monod isometry groups curved spaces structure theory journal topology page caprace michah sageev rank rigidity cat cube complexes geometric functional analysis michael davis groups generated reflections aspherical manifolds covered euclidean space annals mathematics pages matthew day finiteness outer automorphism groups random artin groups algebr geom carl droms isomorphisms graph groups proceedings american mathematical society martin dunwoody accessibility finitely presented groups inventiones mathematicae jingyin huang alex eskin benson farb rigidity higher rank symmetric spaces journal american mathematical society alex eskin rigidity nonuniform lattices higher rank symmetric spaces journal american mathematical society benson farb michael handel commensurations publications gerasimov actions cubings siberian advances mathematics gromov hyperbolic groups essays group theory volume math sci res inst pages springer new york haglund isometries cat cube complexes arxiv preprint haglund finite index subgroups graph products geometriae dedicata mark hagen weak hyperbolicity cube complexes groups journal topology ursula hamenstaedt geometry mapping class groups iii rigidity arxiv preprint mark hagen thomas koberda personal communication july jingyin huang bruce kleiner groups raag arxiv preprint jingyin huang top dimensional quasiflats cat cube complexes arxiv preprint jingyin huang classification artin groups several infinite cases arxiv preprint haglund daniel wise special cube complexes geom funct kim thomas koberda artin subgroups braid groups arxiv preprint kim thomas koberda embedability artin groups geometry topology kim thomas koberda geometry curve graph rightangled artin group international journal algebra computation michael kapovich bruce kleiner bernhard leeb rham decomposition topology bruce kleiner bernhard leeb rigidity symmetric spaces euclidean buildings comptes rendus des imathematics bruce kleiner bernhard leeb groups symmetric spaces communications analysis geometry thomas koberda lemmas applications geometry topology geometry topology dynamics character varieties thomas koberda artin groups generalized isomorphism problem finitely generated subgroups mapping class groups geometric functional analysis michael laurence generating set automorphism group graph group journal london mathematical society george mostow strong rigidity locally symmetric spaces volume princeton university press lee mosher michah sageev kevin whyte trees finite depth trees arxiv preprint panos papasoglu kevin whyte groups infinitely many ends commentarii mathematici helvetici michah sageev ends group pairs curved cube complexes proc london math soc herman servatius carl droms brigitte servatius surface subgroups graph groups proceedings american mathematical society classification artin groups herman servatius automorphisms graph groups journal algebra samuel taylor artin groups embeddings arxiv preprint daniel wise research announcement structure groups quasiconvex hierarchy electron res announc math sci courant institute mathematical science new york university mercer street new york usa address jingyin
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shortened workshop version paper https dec erpre probabilistic programming language program induction alexander marc rishabh nate pushmeet jonathan daniel microsoft research perceptiveio mabrocks risin nkushman pkohli dtartlow jtaylor abstract study machine learning formulations inductive program synthesis given examples synthesize source code maps inputs corresponding outputs key contribution erpre language expressing program synthesis problems erpre model composed specification program representation interpreter describes programs map inputs outputs inference task observe set inputoutput examples infer underlying program erpre model automatically perform inference using four different gradient descent thus erpre model seen defining differentiable interpreter linear program relaxations graphical models discrete satisfiability solving ketch program synthesis system erpre two main benefits first enables rapid exploration range domains program representations interpreter models second separates model specification inference algorithm allowing proper comparisons different approaches inference illustrate value erpre developing several interpreter models performing extensive empirical comparison alternative inference algorithms variety program models knowledge first work compare search program space traditional alternatives key empirical finding constraint solvers dominate gradient descent formulations introduction learning computer programs examples inductive program synthesis ips fundamental problem computer science dating back least summers biermann field produced many successes perhaps visible example flashfill system microsoft excel gulwani gulwani also significant recent interest neural models components resemble computer programs giles joulin mikolov grefenstette graves weston kaiser sutskever reed freitas neelakantan kurach andrychowicz kurach models combine neural networks external memory external computational primitives structure reflects desired algorithmic structure execution however none produce programs output instead program hidden inside controllers composed neural networks decide operations perform learned program understood terms executions produces specific inputs work done author microsoft research workshop neural abstract machines program induction nampi nips barcelona spain outputs outputs execution outputs observed element execution terpret interpreter model interpreter terpret interpreter model inference source code inputs params params inputs unobserved element inputs figure high level view program synthesis task forward execution traditional interpreter forward execution erpre model inference erpre model technique name description fmgd forward marginals gradient descent integer linear programming smt tensorflow gradient descent based approach generalizes approach used kurach gurobi novel linear program relaxation supports gates minka winn translation problem logical formula existential constraints cast erpre model partial program interpreter containing holes source code inferred specification examples ketch ketch table erpre inference algorithms work focus models represent programs simple natural source code hindle kind source code people write two main advantages representing programs source code rather weights neural network controller first source code interpretable resulting models inspected human debugged modified second programming languages designed make easy express algorithms people want write using languages model representation inherit inductive biases lead strong generalization generalizing successfully test data systematically differs training course natural source code likely best representation cases example programming languages designed writing programs classify images reason expect natural source code would impart favorable inductive bias case optimization program space known difficult problem however recent progress neural networks showing possible learn models differentiable computer architectures along success gradient optimization raises question whether gradient descent could powerful new technique searching program space issues motivate main questions work ask whether new models designed specifically synthesize interpretable source code may contain looping branching structures searching program space using gradient descent compares combinatorial search methods traditional ips address first question develop models inspired intermediate representations used compilers like llvm lattner adve trained gradient descent models interact external storage handle control flow explicit statements loops appropriately discretized learned model expressed interpretable source code note two concurrent works adaptive neural compilation bunel differentiable forth riedel implement similar models address second question concerning efficacy gradient descent need way specifying many ips problems gradient based approach compared alternative approaches manner across variety domains alternatives originate rich histories ips programming languages inference discrete graphical models knowledge comparison previously performed benefit formulated context erpre erpre provides means describing execution model turing machine assembly language etc defining program representation interpreter maps inputs outputs using source code parametrisation ruletable param tape copy interpreter model tape var input output tape tape range tape tape tape ruletable ruletable tape copy tape tape figure illustrative example erpre model corresponding factor graph describe toy automaton updates binary tape according previous two entries rule refer long version definition graphical symbols parametrized program erpre description independent particular inference algorithm ips task infer execution model parameters program given execution model pairs inputs outputs overview synthesis task appears fig perform inference erpre automatically compiled intermediate representation fed particular inference algorithm table describes inference algorithms currently supported erpre interpretable source code obtained directly inferred model parameters driving design principle erpre strike subtle balance breadth expression needed precisely capture range execution models restriction expression needed ensure automatic compilation range different tractable erpre language due space give simple example erpre models written full grammar language several longer examples see long version gaunt pre obeys python syntax use python ast library parse compile erpre models model composed param variables define program var variables represent intermediate state computation inputs outputs currently variables must discrete constant sizes erpre supports loops ranges defined statements arrays array indexing functions map discrete set inputs discrete output total constraints imply model converted gated factor graph minka winn details works longer version simple erpre model appears fig model binary tape program writes program rule table specifies state write current position tape conditional pattern appeared two previous tape locations first two tape positions initialized inputs final tape position program output used erpre build much complicated models including turing machine boolean circuits basic block model similar intermediate representation used llvm model constants define maximum quantities like maximum number timesteps program execute maximum number instructions program behavior achieved defining absorbing allowing instruction etc turing machine code state read write head tape move new state wire wire wire logic gate wire regout current block instr regcond blockthen else current line instr instr elif goto jnz regout else instr goto goto next next line blockelse turing machine description invert prepend zero binary decrement perfom bitwise inversion binary string tape right shift symbols insert first cell decrement binary representation tape boolean circuits description controlled shift register full adder adder swap bits wires iff wire perform binary addition two bits including carries perform binary addition numbers basic block description access decrement access element contiguous array decrement elements contiguous array access element linked list assembly description branchaddr branchaddr access decrement table overview benchmark problems grouped execution model illustrate basic structure model parameters model inferred denoted experimental results table gives overview benchmark programs attempt synthesize created erpre descriptions execution model designed three synthesis tasks per model specified examples measure time taken inference techniques listed table synthesize program task results summarized table exception fmgd algorithm perform single run timeout hours task fmgd algorithm run vanilla form optimized form additional heuristics gradient clipping gradient noise entropy bonus kurach aid convergence even heuristics observe several random initializations fmgd algorithm stall uninterpretable local optimum rather finding interpretable discrete program global optimum vanilla case report fraction different random initializations lead globally optimal solution consistent specification also wall clock time epochs gradient descent algorithm typical number required reach convergence successful run optimized fmgd case randomly draw sets hyperparameters manually chosen distribution run learning different random initializations setting report success rate best hyperparameters found also average across runs random search results show traditional techniques employing constraint solvers smt sketch outperform methods fmgd ilp ketch system able solve benchmarks furthermore table highlights precise formulation interpreter model affect speed synthesis basic block assembly models equally expressive assembly model biased towards producing straight line code minimal branching cases synthesis successful assembly representation seen outperform basic block model terms synthesis time addition performed separate experiment investigate local optima arising fmgd formulation using erpre describe task inferring values bit string length observation parity neighboring variables possible show analytically number local optima grow exponentially table provides empirical evidence minima encountered practice hinder convergence fmgd algorithm additional pre processing performed ketch makes slower raw smt solver smaller tasks allows succeed larger tasks conclusion presented erpre probabilistic programming language specifying ips problems flexibility erpre language combination four inference backends allows comparison techniques inductive program synthesis primary experiments constraint solvers outperform approaches cases studied however work intentionally measuring ability efficiently search program space remain optimistic extensions erpre framework allow differentiable interpreters handle problems involving perceptual data gaunt using machine learning techniques guide techniques balog references marcin andrychowicz karol kurach learning efficient algorithms hierarchical attentive memory arxiv preprint matej balog alexander gaunt marc brockschmidt sebastian nowozin daniel tarlow deepcoder learning write programs arxiv preprint alan biermann inference regular lisp programs examples ieee transactions systems man cybernetics rudy bunel alban desmaison pushmeet kohli philip torr pawan kumar adaptive neural compilation corr url http alexander gaunt marc brockschmidt nate kushman daniel tarlow lifelong perceptual programming example arxiv preprint alexander gaunt marc brockschmidt rishabh singh nate kushman pushmeet kohli jonathan taylor daniel tarlow terpret probabilistic programming language program induction arxiv preprint lee giles sun chen lee dong chen higher order recurrent networks grammatical inference advances neural information processing systems nips conference denver colorado usa november pages alex graves greg wayne ivo danihelka neural turing machines corr url http edward grefenstette karl moritz hermann mustafa suleyman phil blunsom learning transduce unbounded memory advances neural information processing systems pages sumit gulwani automating string processing spreadsheets using examples acm sigplan notices volume pages acm sumit gulwani william harris rishabh singh spreadsheet data manipulation using examples communications acm aug abram hindle earl barr zhendong mark gabel premkumar devanbu naturalness software international conference software engineering icse pages ieee armand joulin tomas mikolov inferring algorithmic patterns recurrent nets advances neural information processing systems nips conference denver colorado usa november pages kaiser ilya sutskever neural gpus learn algorithms proceedings international conference learning karol kurach marcin andrychowicz ilya sutskever neural machines proceedings international conference learning representations url http time vanilla fmgd best hypers average hypers ilp time smt time ketch time invert prepend zero binary decrement boolean circuits controlled shift register full adder adder basic block access decrement assembly access decrement turing machine table benchmark results fmgd present time seconds epochs success rate random restarts vanilla best hypers average hypers columns respectively present time seconds produce synthesized program symbol indicates timeout failure random restart converge number provided examples used specify task case vanilla fmgd best hypers average hypers table percentage runs converge global optimum fmgd parity chain example length chris lattner vikram adve llvm compilation framework lifelong program analysis transformation code generation optimization cgo international symposium pages ieee tom minka john winn gates advances neural information processing systems pages arvind neelakantan quoc ilya sutskever neural programmer inducing latent programs gradient descent proceedings international conference learning representations scott reed nando freitas neural proceedings international conference learning representations sebastian riedel matko bosnjak tim programming differentiable forth interpreter corr url http armando program synthesis sketching phd thesis eecs berkeley phillip summers methodology lisp program construction examples journal acm jacm jason weston sumit chopra antoine bordes memory networks proceedings international conference learning representations url http
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classification patient notes case study icd code assignment tal baumel nov university israel jumana raphael cohen michael elhadad elhadad san francisco university israel columbia university new york university israel abstract automatic coding clinical documentation according diagnosis codes useful task electronic health record challenging one due large number codes length patient notes investigate four models assigning multiple icd codes discharge summaries experiment data mimic iii clinical datasets present hierarchical attentionbidirectional gated recurrent unit hierarchical approach tag document identifying sentences relevant label achieves art results furthermore learned attention layer highlights model decision process allows easier error analysis suggests future directions improvement introduction electronic health records ehrs often need assign multiple labels patient record choosing large number potential labels diagnosis code assignment task massive amount labels chose codes codes largescale multiple phenotyping assignment problem list identification even intermediate patient representation cast classification large label set recently context predictive modeling approaches predict multiple future healthcare outcomes future diagnosis codes medication orders proposed literature setup occurs data fed classification large label set paper investigate leverage unstructured portion ehr patient notes along novel application neural architectures focus three characteristics large label set unique codes unique codes setting labels per instance iii instances long documents discharge summaries average long furthermore work long documents one critical aspect classification highlight elements documents explain support predicted labels much work characteristics copyright association advancement artificial intelligence rights reserved limited work tackle particularly clinical domain experiment four approaches classification model continuous cbow model convolutional neural network cnn model bidirectional gated recurrent unit model hierarchical attention mechanism among attention mechanism model provides full transparency classification decisions rely publicly available mimic datasets validate experiments characteristic healthcare domain long documents large number technical words experiment simple yet effective preprocessing input texts results show careful tokenization input texts hierarchical segmentation original document allow hierarchical attention gru architecture yield promising results svm cbow cnn models preserving full input text providing effective transparency previous work review previous work healthcare domain well recent approaches extreme classification take place range domains tasks patient classifications approaches classification patient records multiple labels fall three types tasks diagnosis code assignment patient record labeling predictive modeling diagnosis code assignment automated icd coding well established task several methods proposed literature ranging rule based crammer farkas szarvas machine learning support vector machines bayesian ridge regression knearest neighbor larkey croft lita methods exploit hierarchical structure icd taxonomy perotte perotte others incorporated explicit relations codes kavuluru rios many cases handle sheer amount labels different approaches focus icd codes version codes descendants icd taxonomy subset codes like shared community task radiology code assignment pestian difficult compare different methods proposed since relies different usually publicly available datasets experiment mimic dataset since publicly available research community methodswise approach departs previous work two important ways experiment massively large large label sets code codes experiment transparent models highlight portions input text support assigned codes patient record labeling automated diagnosis coding patient record classifiers fall tasks phenotyping across multiple conditions instance uphenome model takes probabilistic generative approach assign latent variables pivovarov recently context learning harutyunyan colleagues experimented phenotyping critical care conditions harutyunyan predictive modeling previous work ehr classification mostly focused predictive scenarios size label set varies one approach another limit label set size however deeppatient miotto predicts set condition codes lipton leverage lstm model predict vocabulary diagnosis codes doctorai choi bahadori sun predicts set icd codes medication groups survival filter ranganath predicts series future icd codes across approximately icd codes inputs classifications work classification takes structured input instance survival filter expects icd codes input predict future icd codes doctorai takes input medication orders icd codes problem list procedure orders given visit deep patient take content notes input content heavily preprocessed structured input neural network tagging texts medical named entities contrast approach leverage entire content input texts work contributes clinical natural language processing demner fushman elhadad recently investigated neural representations architectures traditional tasks named entity recognition jagannatha extreme classification extreme learning objective annotate data point relevant subset labels extremely large label set much work carried outside healthcare domain tasks image classification tsoumakas katakis weston bengio usunier question answering choi advertising jain prabhu varma weston bengio usunier task annotating large dataset images large label set first addressed authors introduced wsabie method relies two main features records images labels embedded shared lowdimension vector space classification task modeled ranking problem evaluated hamming loss metric proposed online approximate warp loss allowed algorithm perform fast enough scale dataset found case standard measure appropriate tolerate approximate annotations extent image annotation task sleec method bhatia also relies learning embedding transformation map label vectors representation sleec learns ensemble local distance preserving embeddings accurately predict infrequently occurring labels approach attempts exploit similarity among labels improve classification learns different representations clusters similar labels approaches attempt reduce cost training large datasets considering part labels classification decision yen sleec later improved jain prabhu varma pfastrexml method also adopted loss functions aiming predicting tail labels joulin fasttext method introduced simple scalable neural bag words approach assigning multiple labels text test similar model cbow experiments one baselines dataset preprocessing use publicly available mimic dataset icu stays beth israel deaconess medical center saeed johnson mimic datasets test impact training size relied mimic mimic iii datasets mimic iii comprises records collected described expansion mimic comprises records collected along edits dataset including procedures compare experiments previous work icd coding used publicly available split mimic perotte contains discharge summaries divided training set summaries testset unseen patients summaries thus kept train mimic constructed additional training set mimic iii made sure patients remained unseen training set well overall two training sets refer mimic mimic iii common comprising summaries unseen patients large overlap mimic mimic iii also marked differences found records unique tokens avg tokens record avg sentences record full labels labels label cardinality label density labels records mimic mimic iii test set table datasets descriptive statistics many cases discharge summaries found one dataset addition mimic iii contains addenda discharge summaries part mimic examining summaries addenda noticed addenda contain vital information icd coding missing main discharge summaries therefore decided concatenate summaries addenda table reports descriptives statistics regarding datasets overall mimic iii larger mimic standpoints including amounts training data vocabulary size overall number labels codes label set comes taxonomy international classification diseases icd repository maintained world health organization provide standardized system diagnostic codes classifying diseases hierarchical structure connecting specific diagnostic codes relations hierarchy eight levels less specific specific icd codes contain diagnosis procedure codes paper focus diagnosis codes codes conveyed digits primary digits secondary ones table provides label cardinality density defined tsoumakas katakis cardinality average number codes assigned records dataset density cardinality divided total number codes training sets number labels order number records label density extremely low confirms task code assignment belongs family extreme classification filter icd code based frequency note however approximately frequent labels defined assigned least records table experimented two versions label set one labels one labels rolled equivalent input texts tokenization preprocessing input records comprised following steps tokenize input texts ing spacy library convert characters mapped iii build vocabulary tokens appear least times training set map word nearest word vocabulary using edit distance step simple yet particularly useful reducing number misspellings medical terms preprocessing steps strong impact vocabulary instance unique tokens mimic iii test set preprocessing remaining vocabulary preprocessing drop step improved performance tested cbow cnn methods reported hierarchical segmentation besides tokenization input texts carried one level segmentation sentence level using spacy library well two reasons preprocessing input texts sentence segmentation first deal long documents impossible ineffective train sequence model like gru long sequences previous approaches document classification problem resolved truncating input documents case discharge summaries however acceptable solution want preserve entire document transparency second inspired moving windows johnson zhang posit sentences form linguistically inspired windows word sequences beyond tokens sentences discharge summaries exhibit strong structure history present illness past medical history followed hospital course discharge plans lipsky gorman elhadad presents exciting opportunity future work exploit discourse segments additional representation layer input texts methods describe four models experimented icd coding evaluated literature according different metrics variant takes account hierarchy codes perotte hamming ranking loss wang modified version mean reciprocal rank mrr subotin davis evaluate performance using microf metric since commonly used metric svm used scikit learn pedregosa implement binary svm classifier features bag words idf weights determined corpus release notes label stop words removed using scikit learn default english list model fits binary svm classifier label icd code rest labels also experimented feature filtering select words https binary loss binary loss fully connected sigmoid activation fully connected sigmoid activation max pooling average convolution layer embedding embedding diagnoses share lexical words distinguished model cnn address problems cbow model next model investigate convolutional neural network cnn one dimensional convolution applied list embedded words could considered type model convolution filter size architecture model similar cbow model instead averaging embedded words apply one dimensional convolution layer filter followed max pooling layer output max pool layered fully connected layer applied like cbow model also experimented deeper convolution networks inception module lecun yield improved results embedding figure cbow architecture left cnn model architecture right according mutual information label improve performance cbow cbow model inspired cbow model mikolov fasttext joulin methods use simple create dense representation words use average representation prediction cbow tries predict word words appear around cbow model icd classification predicts codes words input discharge summary model architecture consists embedding layer applied words given input text encoding vector vocabulary embedding matrix dimension nemb size vocabulary nemb embedding size set embedded words averaged vector fed fully connected layer matrix bias output dimension number labels use sigmoid activation output layer values range use fixed threshold determine whether assign particular label train model used binary loss loss target output target log output target log output embedding averaged rob sigmoid averaged model extremely lightweight fast suffers known issues ignores word order negation appear diagnosis mention model would able learn identified model different conved max embedding rob sigmoid conved experiments used embedding parameter cbow model addition set number channels filter size introduce hierarchical attentionbidirectional gated recurrent unit model adaptation hierarchical attention networks yang able handle classification gated recurrent unit gru type recurrent neural network since documents long see table tokens mimic iii training set regular gru applied entire document slow requires number layers document length instead apply hierarchal model two levels bidirectional gru encoding first bidirectional gru operates tokens encodes sentences second bidirectional gru encodes document applied encoded sentences architecture gru applied much shorter sequence compared single gru take advantage property label invoked different parts text use attention mechanism second gru different weights label allows model focus relevant sentences label choi allow clarity model learns enable error analysis attention also applied first gru weights labels sentence input text encoded fixed length vector applying embedding layer inputs applying bidirectional gru layer embedded words using neural attention mechanism encode bidirectional gru outputs size sentences encoded fixed length vector apply second bidirectional gru layer sentences using different attention layers generate encoding specified class labels finally applied fully connected layer softmax classifier determine label assigned document training codes mimic mimic iii classifiers classifiers classifiers svm cbow cnn document gru layer sentence encoder sentence encoder sentence encoder codes mimic mimic iii table two settings full icds four models trained mimic mimic iii datasets figure model architecture overview results achieved using categorical every classifier separately loss target output ouput log target attw eight ini tanh ini attw eight ini eattw eight ini attw eightj attend sum ini attw eight ini embedding encsentsj attend gru words embedding vwords wwords encdoclabel attend grusents encsents vlabel wlabel roblabel sof tmax pwlabel encdoclabel pblabel encoding vector vocabulary size embedding matrix size nemb gruwords gru layer state size hstate wwords square matrix hstate hstate vwords vector hstate sentence level attention grusents gru layer state size hstate wlabel square matrix hstate hstate vlabel vector hstate document level attention class pwlabel matrix hstate pblabel bias vector size label implemented model using dynet neubig softmax fully connected layer classifier embedding sentence encoder sentence gru layer evaluate proposed methods mimic datasets conducted following experiments first setting considered codes label set trained svm cbow cnn mimic mimic iii training sets separately models evaluated test set according second setting considered codes codes table gives best results setting improvement cnn svm second best methods mimic mimic iii respectively full scenario methods yield better results trained mimic iii rather mimic expected considering larger size mimic iii note cnn yields best trained mimic iii passing small margin comparison previous work perotte svm yielded better results flat hierarchy classifiers trend confirmed training new mimic iii set well using evaluation metrics perotte attribute improved results approach well tokenization approach label frequency categorical crossentropy loss weighted average sentence attention model comparison weighted average label attention figure sentence encoder classifier also tested effect label frequency performance classifier recalculated precision recall scores subsets labels subsets created sorting labels frequency appear dataset binning groups labels bin comprises frequent codes training set average frequency records training set codes bin average frequency codes bin appeared records bin records training set effect seen figure note recall score drops much dramatically precision label frequency decreases model explaining power code available https mimic discuss cnn architectures support model explaining power figure sample text patient note one sentence per line left visualization attention weights sentence word levels associated codes left sentence level attention weights code heart failure right code traumatic pneumothorax hemothorax find word received highest attention score example experiments label failure found sentence highest attention score congestive heart failure ejection fraction token failure found relevant across labels figure provides additional examples note tokens step mapped numbers pseudotokens figure effect label frequency performance trained mimic iii represents bins labels ranked frequency training set cnn analyze cnn prediction test triggered layer given sentence words feed forward embedding layer convolution layer output convolution list vectors size number channels convolution layer vector corresponds identify triggered max pooling layer finding maximum value channel thus predicted labels one activated include information relevant label whether correct true positive labels incorrect false positive labels example experiments label abscess leg except foot one activated detected extremity cellulitis prior transparency process also useful error analysis building model highlight true positive false positive labels however difficult cnn trace back decisions false negatives predictions model use attention weights better understand sentences words sentence contributed decision find sentence highest attention score label given important sentence like cnn use process error analysis fact model explains prediction greater precision sentence level instance could explore following false positive prediction model assigned label cerebral degenerations sentence alzheimer dementia see condition relevant medical note mentioned patient past medical history current problem fact many false positive labels model due mentions belonging past medical history section suggests coding task would benefit deeper architecture attention structure contrast cnn model also help analyze false negative label assignments explored false negative labels found many cases model found relevant sentence failed classify correctly suggests attention mechanism successful instance false negative cellulitis abscess attended sentence right lower extremity cellulitis prior admission false positive codes sentence included mellitus forms chronic ischemic heart disease note case cellulitis reasonable classifier preferred frequent codes common comorbid condition full visualizations sample discharge summaries provided https mimicdemo conclusion investigate four modern models task extreme classification mimic datasets unlike previous work evaluate models codes thus making sure models could used real world tagging tokenization step mapping rare variants using edit distance improved results cbow cnn models highlighting importance preprocessing data noise problems settings model achieves best performance task codes mimic iii absolute improvement best svm baseline able provide insight task future work using structure available medical notes yet never used ability highlight decision process model important adoption models medical experts mimic includes smaller training dataset achieved absolute improvement suggesting requires less training data achieve top performance important domain adaptation efforts applying models patient records sources different hospitals acknowledgements work supported national institute general medical sciences grant frankel center computer science references bhatia bhatia jain kar varma jain sparse local embeddings extreme classification advances neural information processing systems nips choi bahadori sun choi bahadori sun doctor predicting clinical events via recurrent neural networks arxiv preprint choi choi hewlett lacoste polosukhin uszkoreit berant hierarchical question answering long documents arxiv preprint crammer crammer dredze ganchev talukdar carroll automatic code assignment medical text proceedings acl workshop bionlp biological translational clinical language processing demner fushman elhadad demner 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dudley deep patient unsupervised tation predict future patients electronic health records scientific reports neubig neubig dyer goldberg matthews ammar anastasopoulos ballesteros chiang clothiaux cohn dynet dynamic neural network toolkit arxiv preprint pedregosa pedregosa varoquaux gramfort michel thirion grisel blondel prettenhofer weiss dubourg scikitlearn machine learning python journal machine learning research oct perotte perotte wood elhadad bartlett hierarchically supervised latent dirichlet allocation advances neural information processing systems nips perotte perotte pivovarov natarajan weiskopf wood elhadad diagnosis code assignment models evaluation metrics journal american medical informatics association pestian pestian brew matykiewicz hovermale johnson cohen duch shared task involving classification clinical free text proceedings acl workshop bionlp biological translational clinical language processing pivovarov pivovarov perotte grave angiolillo wiggins elhadad learning 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document classification proceedings yen yen huang zhong ravikumar dhillon primal dual sparse approach extreme multiclass multilabel classification proceedings international conference machine learning icml
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submitted brazilian journal probability statistics arxiv searching core variables principal components analysis jan yanina gimeneza guido giussania universidad san conicet abstract article introduce procedure selecting variables principal components analysis developed identify small subset original variables best explain principal components nonparametric relationships usually noisy uninformative variables dataset variables strongly related one another general dependence procedure designed used following satisfactory initial principal components analysis variables aim help interpret underlying structures analyze asymptotic behavior method provide examples introduction principal components analysis pca best known dimensional reduction procedure multivariate data important drawback pca sometimes provides poor quality interpretation data practical problems final output linear combination original variables aim present study identify small subset original variables dataset whilst retaining information related first principal components large body literature focuses trying interpret principal components jolliffe introduced rotation techniques vines proposed restrict value loadings pca small set allowable integers mccabe presented different strategy aims select subset original variables similar criterion pca years later whole literature variable selection appeared inspired lasso least absolute shrinkage selection operator technique introduced tibshirani way lasso works described thus shrinks coefficients sets others hence tries retain good features subset selection ridge primary secondary keywords phrases informative variables multivariate analysis principal components selection variables gimenez giussani regression lasso minimizes residual sum squares subject sum absolute value coefficients less constant nature constraint tends produce coefficients exactly hence gives interpretable jolliffe proposed scotlass imposes bound sum absolute values loadings component using similar idea one lasso used regression zou presented spca sparse pca extends elastic net zou hastie generalization lasso mentioned spca built fact pca written optimization problem quadratic penalty lasso penalty via elastic net directly integrated regression criterion leading modified pca sparse luss aspremont studied application sparse pca clustering problems feature selection sparse pca seeks sparse factors linear combinations variables dataset explain much variance data possible limiting number nonzero coefficients far possible authors applied results classic biological clustering feature selection problems recently witten tibshirani introduced notion lassoed principal components identifying genes considered problem testing significance features high dimensional data approach rather different designed used satisfactory pca achieved rather methods produce principal components particular characteristics coefficients zero interpretable principal components produced first perform classical pca look small subset original variables contain almost relevant information explain principal components however method also used performing sparse pca method described previously develop method borrowed ideas selecting variables fraiman introduced two procedures selecting variables cluster analysis classification rules procedures based idea blinding unnecessary variables cancel effect variable substituted values marginal mean first procedure conditional mean second marginal mean approach mainly intended identify noisy uninformative variables conditional mean approach could also deal dependence adapted idea behind second procedure pca searching core variables principal components analysis section introduce main definitions population version proposed method empirical version also present main results section simulation study conducted results compared pca variable selection procedures finally section study real data example proofs given appendix method definitions properties begin defining notation stating problem terms underlying distribution random vector give estimates based sample data via empirical distribution population version define random vector distribution coordinates vector defined covariance matrix denoted given random vector say satisfy assumption covariance matrix positive definite iii eigenvalues covariance matrix different throughout manuscript fulfills assumption well known first principal component associated vector defined arg max arg max next principal components defined arg max arg max subspace generated vectors spectral theorem follows eigenvalues solutions pca corresponding eigenvectors gimenez giussani given subset indices cardinality define subset random variables slight abuse notation also denote vector define vector depends variables principal components associated vector depend variables distribution denoted pyi covariance matrix regression function follows assume fulfills assumption remark assumptions hold variable linear combination variables instance gaussian independent variables practice see problem tackled example section looking subset small cardinality minimizes distance original principal components principal components function variables subset done following two different approaches local approach define objective function pyi pyi measures squared distance first original principal component first principal component function variables subset given fixed integer family subsets cardinality family subsets minimum attained equivalently analogously define pyi pyi searching core variables principal components analysis measures squared distance original principal component principal component function variables subset principal components may cases rather difficult interpret important issue practice find small cardinal subset objective function small enough principal component well explained subset original variables global approach case objective function define time objective function deals finding unique subset explain first principal components think components equally important choose otherwise components important others put different weights default suggest choosing weights proportional variance component explaining local approach consider different subset principal component using objective function global approach seek unique subset first principal components using practice choose first principal components explain high percent total variance consider subset principal components subsets tell original variables best explain first principal components follows refer method blinding procedure important issue choose cardinality set one hand objective function decreases gimenez giussani increases every hand look subset small cardinality constant tug war since components unitary norms direct relationship angle two vectors clear smaller angle closer components hence component propose fix angle choose smallest value makes angle blinding original component smaller want explain first principal components propose fix angle choose smallest value makes largest angles smaller cases angle must chosen user default propose fix angle larger degrees empirical version consistent estimates optimal subset aimed consistently estimate sets sample random vectors distribution given subset first step build sample yni random vectors depends using nonparametric estimates conditional expectation regression function assume assumption gni strongly consistent estimate uniformly conditions holds found hansen first define empirical version blinded observations example consider nearest neighbours estimator therefore set integer value number nearest neighbours going use respect appropriate metric typically euclidean mahalanobis distance considering coordinates precisely find set indices neighbours among next define random vectors yji verifying yji otherwise stands vector searching core variables principal components analysis corresponds use nonparametric estimate gni gni since gni strongly consistent estimate moreover observe notation emphasize function depends coordinates indicated subset stands empirical distribution associated stands empirical distribution yji examples consistently estimate optimal number nearest neighbours use generalized cross validation procedure proposed gong rbopt arg min large nonparametric estimators perform poorly due curse dimensionality case approach used like proposed shi also recent proposal biau called cobra used avoid curse dimensionality consistency result given following theorem still valid long estimates verify consistency assumptions required purely nonparametric estimates mainly interested cases small experience good idea start search genetic algorithm provides initial solution variables small improve result working card also algorithm one proposed fraiman used finally define corresponding empirical versions respectively gimenez giussani let observe corresponds principal component associated sample yji corresponds objective function principal component measures square distance original principal component principal component function variables subset function determines variables retain information related principal component variables choosing ones minimize function subset indicates variables case looking unique subset explain first principal components consider function robust version obtained using robust principal components see instance maronna replacing local mean local median theorem assumptions ultimately probability one also ultimately proof proof given appendix simulated examples section consider two simulated experiments analyze behavior method compare methods proposed literature example better understand heart procedure start simple simulation example dimension four two hidden factors independent searching core variables principal components analysis construct observable variables follows case case case perform variable selection clear two variables containing information given kept means would good choices clearly bad choice since retains information additionally respec good choice since recover information respec example cases compare procedure proposals algorithms jolliffe variable selection approach proposed mccabe sparce pca introduced zou retain variables algorithm associates one original variables last pca vectors deletes variables associates one original variables first pca vectors retains variables case perform replicates generate samples size model compute covariance matrix case first two principal components explain percent total variance times maximum angle conformed first two principal components smaller degrees hence keep two variables results exhibitted table page show times makes good choice variables times methods fail replications case first two principal components explain percent total variance times maximum angle conformed first two principal components smaller degrees hence keep two variables results exhibitted table page show times makes good choice variables times spca mccabe fails replications gimenez giussani table proportion times method selects pair variables case mccabe spca table proportion times method selects pair variables case mccabe spca case first two principal components explain percent total variance times maximum angle conformed first two principal components smaller degrees hence keep two variables results exhibitted table page show times makes good choice variables times spca mccabe table proportion times method selects pair variables case mccabe spca definition function plus error see table page table page table page procedure selects proportion times procedures select around proportion times cases procedure detects function selected one two explanatory variables one select searching core variables principal components analysis way gains information instead getting redundant information choosing note also good choice make problem bit challenging enlarge dimension dataset repeating variables plus noisy noninformative errors also add noisy variables precisely consider following model case case case keep first second principal component replicates explain total variance case case case require methods select two variables three cases least times largest angle smaller degrees hence consider good decision look two variables five groups variables within group variables differ noisy noninformative error precisely first four groups last one clearly bad choices also good choices either table page table page table page exhibit proportion replicates method selects one variable per group variables three cases achieves desired results times methods fail times example zou introduced following simulation example two hidden factors gimenez giussani table proportion replicates method selects one variable per group variables mccabe spca table proportion replicates method selects one variable per group variables mccabe spca lineal combination independent observable variables constructed follows searching core variables principal components analysis table proportion replicates method selects one variable per group variables mccabe spca zou used true covariance matrix perform pca spca simple thresholding three groups variables share information first group denote corresponds variables second one variables third group variables definition also function noisy noninformative error zou show spca correctly identifies sets important variables first spca identifies group variables second spca group variables simple thresholding incorrectly mixes two variables two first consider population version using true covariance matrix first two principal components explain total variance example outside theoretical framework stated section problem implemetation procedure clearly adecuate keep one variable cardinality one positive eigenvalue gimenez giussani recover two principal components choose subset cardinal two positive eigenvalues multiplicity one recover two principal components select one variable three groups good solution bad solution choose variables within group table theoretical objective value two variables chosen one group table page see takes large values larger one bad choices small values smaller good choices moreover good choices largest angle original blinded components less two dergees cases going select good choice method performs perfectly well cardinal next consider realistic situation perform small simulation estimate covariance matrix generating sample size iterate times perform analysis considering two first principal components explain variance apply procedure select two variables first two principal components cases larger angle degrees cases larger degrees table page see times makes good choice addition mccabe spca always make good choice times makes good choice times real data example consider dataset obtained university california irvine repository frank asuncion example dataset contains values six characteristic used classify orthopaedic patients three classes normal disc hernia spondylolisthesis groups includes data one hundred normal patients sixty searching core variables principal components analysis table proportion times method selects one variable per group variables mccabe spca patients disc hernia one hundred fifty patients spondylolisthesis patient represented dataset terms six attributes associated shape orientation pelvis lumbar spine namely pelvic incidence pelvic tilt lumbar lordosis angle sacral slope pelvic radius grade spondylolisthesis use procedure select one variable first two principal components explain variance give weight components equation grade spondylolisthesis variable selected case largest angle degrees hence decide keep one variable finds nearest neighbours variables variable variable variable empirical objective function attains value use euclidean distance procedure mahalanobis distance get results second component second component first component first component figure left map projection data original principal components right map projection data blinded based principal components blue disk hernia patients black normal patients green spondylolisthesis patients gimenez giussani second component second component second component first component first component first component figure left map projection data original principal components middle map projection data blinded based principal components using euclidean distance right map projection data blinded based principal components using mahalanobis distance blue disk hernia patients black normal patients plot first two principal components example data looks similar plot principal components calculated using procedure figure also shows patients spondylolisthesis separated rest patients normal patients patients disc hernias mixed together next calculate two principal components using traditional method procedure subset normal disc hernia patients using two variables first two principal components explain variance decide keep discard select one variable case largest angle close degrees two variables largest angle close degrees three variables angle decrease much degrees decide keep two variables consider euclidean mahalanobis distance cases variables selected lumbar lordosis angle pelvic radius value objective function euclidean distance mahalanobis distance figure shows plots principal components produced procedures look similar conclusions paper introduce new variable selection procedure pca aim gain interpretation principal components one hand consider local approach deals finding subset variables best explains principal component hand analyzed global approach objective find searching core variables principal components analysis unique subset explain first principal components explained conditional expectation choose variables wide regular conditions conditional expectation estimates covariance matrix strong consistency results studied numerical aspects also analysed performance procedure compared several wellknown variable selection techniques using real simulated data sets showing strengths method appendix proof theorem first going proof following proposition proposition converges uniformly arg min arg min proof since discrete finite space convergence uniform sake simplicity let proof analogue exists every probability one know exists gimenez giussani choosing obtain probability one means exists probability one choose minimizes moreover fix exists let probability one means exists minimizes probability one conclusion max minimizes minimizes probability one proof analogue implies arg min arg min searching core variables principal components analysis proof theorem prove theorem enough see empirical objective function converges theoretical objective function prove let see pyi dauxois proved max respec eigenvectors denotes empirical covariance matrix associated respec denotes covariance matrix associated prove max pyi respec pyi eigenvectors denotes empirical covariance matrix associated respec denotes covariance matrix associated pyi note classic case show also holds max simplify notation assume without losing generality gni uniformly consistent nonparametric estimate specifically gni fulfil following assumption gnl uniformly gimenez giussani define non observable auxiliary vector proof complete show max max considering gnl three matrices cov cov cov cov gni gni prove convergence first show max searching core variables principal components analysis sufficient prove coordinates matrix converge zero set using finite dimensional space coordinates converge zero holds max show max going prove coordinates matrix converge zero better understanding define gnl gnl gni gni inequality hand gimenez giussani addition since gnl uniformly gnl max gnl therefore implies thus required proof simetry matrices gni gni add subtract gni rearrange get sum gni searching core variables principal components analysis rewrited triangular inequality hand gimenez giussani entails conclude proved coordinates matrix converge zero sup ready complete proof theorem able use result dauxois derive pyi indeed pyi pyi pyi pyi pyi pyi pyi pyi entails finally proposition implies arg min arg min searching core variables principal components analysis concludes proof acknowledgements work partially supported grant pict anpcyt agencia nacional argentina grateful ricardo fraiman invaluable assistance dra marcela svarc helpful suggestions authors would like thank referees constructive comments improve significantly work references biau fischer guedj malley cobra collective regression strategy dauxois pousse romain asymptotic theory principal component analysis vector random function applications statistical inference journal multivariate analysis fraiman justel svarc selection variables cluster analysis classification rules journal american statistical association frank asuncion uci machine learning repository http irvine university california school information computer science hansen uniform convergence rates kernel estimation dependent data econometric theory shi bivariate partly linear models journal multivariate analysis jolliffe rotation principal components choise normalization constraints journal applied jolliffe principal components analysis second edition springer jolliffe trendafilov uddin modified principal component technique based lasso journal computational graphical statistics gong nonparametric estimation regression functions presence irrelevant variables econometrics journal luss aspremont clustering feature selection using sparse principal component analysis optimization engineering maronna martin yohai robust statistics theory methods wiley london mccabe principal variables technometrics gimenez giussani tibshirani regression shrinkage selection via lasso journal royal statistical society series vines simple principal components applied statistics witten tibshirani testing significance features lassoed principal components annals applied statistics zou hastie regularizations variable selection via elastic net journal royal statistical society series zou hastie tibshirani sparse principal component analysis journal computational graphical statistics universidad san vito dumas victoria buenos aires argentina conicet yanugimenez ggiussani
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pyramidal evolutionary algorithm different partnering strategies scheduling problems gecco proceedings genetic evolutionary computation conference latebreaking papers volume san francisco usa uwe aickelin school computer science university nottingham uxa abstract paper combines idea hierarchical distributed genetic algorithm different interagent partnering strategies cascading clusters built bottom optimising larger parts problem hence subpopulations search larger search space lower resolution whilst subpopulations search smaller search space higher resolution effects different partner selection schemes amongst agents solution quality examined two multiplechoice optimisation problems shown partnering strategies exploit problemspecific knowledge superior counter inappropriate fitness measurements introduction hierarchically distributed evolutionary algorithms combined structures number new questions become apparent one questions addressed paper issue intelligently selecting mating partners another population agent paper look seven different partnering strategies combined genetic algorithm uses structure evaluate different strategies according optimisation performance two scheduling problems genetic algorithms generally attributed holland students although evolutionary computation dates back refer fogel extensive review early approaches genetic algorithms stochastic metaheuristics mimic features natural evolution canonical genetic algorithms intended function optimisation discussed jong however slightly modified versions proved successful introduction genetic algorithms function optimisation see deb twist applying type distributed genetic algorithm lies special hierarchical structure subpopulations follow different fitness functions effect searching specific parts solution space following special parts gradually merged full solutions advantage divide conquer approach reduced epistasis within makes optimisation task easier genetic algorithm paper arranged follows following section describes nurse scheduling tenant selection problems pyramidal genetic algorithms application two problems detailed section section explains seven partnering strategies examined paper section describes use computational results final section discusses findings draws conclusions nurse scheduling tenant selection problems two optimisation problems considered paper nurse scheduling problem tenant selection problem number characteristics make ideal testbed enhanced genetic algorithm using partnering strategies firstly class problems johnson martello toth hence challenging problems secondly proved resilient optimisation standard genetic algorithm good solutions found using novel strategy indirectly optimising problem decoder based genetic algorithm aickelin dowsland finally problems similar allocation problems nurse scheduling choice allocate nurse whilst tenant selection allocate area mall shop however following detailed explanation two show two problems also distinct characteristics making different yet similar enough interesting comparison results problem creating weekly schedules wards nurses major hospital schedules satisfy working contracts meet demand given numbers nurses different grades shift whilst time seen fair staff concerned latter objective achieved meeting many nurses requests possible considering historical information ensure unsatisfied requests unpopular shifts evenly distributed due various hospital policies nurse normally work subset total theoretically possible instance nurse work either days nights given week interested reader directed aickelin dowsland dowsland details problem purposes problem modelled follows nurses scheduled weekly ward basis work feasible pattern regards contract demand days nights qualification levels covered total three qualification levels corresponding demand exists hospital policy qualified nurses allowed cover less qualified one infeasible solutions respect cover acceptable solution problem would string number elements equal number nurses element would indicate worked particular nurse depending nurses preferences recent history patterns worked overall attractiveness pattern penalty cost allocated pair values set close consultation hospital range perfect unacceptable bias lower values sum values gives quality schedule data sets available average problem size nurses per ward possible per nurse comparison data sets solved using standard package fuller however remained unsolved allowed hours pentium experiments number descent methods using different neighbourhoods standard simulated annealing implementation even less successful frequently failed find feasible solutions successful approach date based tabu search dowsland however quality solutions relies heavily chains moves work well way different factors affecting quality schedule combined straightforward genetic algorithm approach failed solve problem aickelin dowsland best evolutionary results date achieved indirect genetic approach employing decoder function aickelin dowsland however believe leverage direct evolutionary approaches problem hence propose use enhanced pyramidal genetic algorithm paper second problem mall layout tenant selection problem future mall problem short mall problem arises planning phase new shopping centre completion type number shops occupying mall decided maximise revenue good mixture shops heterogeneous homogeneous achieved due difficulty obtaining data confidentiality problem data used research constructed artificially closely modelled actual problem described instance bean following briefly outline model objective mall problem maximise rent revenue mall although small fixed rent per shop large part shop rent depends sales revenue generated therefore important select right number size type tenants place right locations maximise revenue outlined bean rent shop depends following factors attractiveness area shop located total number shops type mall size shop possible synergy effects neighbouring similar shops shops group used bean fixed amount rent based type shop area located problem modelled follows placing shops mall divided discrete number locations big enough hold smallest shop size larger sizes created placing shop type adjacent locations hence problem placing menswear locations belong one groups clothes shops location situated one areas type shop minimum ideal maximum number allowed mall consumers drawn mall balance variety homogeneity shops size shops determined many locations occupy within area purpose study shops grouped three size classes namely small medium large occupying one two three locations one area mall respectively instance two locations filled within one area shop medium size five locations assigned area form one large one medium shop etc usually also maximum total number small medium large shops allowed mall test robustness performance algorithms thoroughly problem problem instances created problem instances locations grouped five areas however sets differ number available tightness constraints regarding minimum maximum number shops certain type size full details data created dimensions differences sets found aickelin pyramidal genetic algorithms problems failed optimised standard genetic algorithm aickelin dowsland previous research showed difficulties attributable epistasis created constrained nature optimisation briefly epistasis refers nonlinearity solution string davidor individual variable values good right particular shift location particular nurse shop formed low quality solutions combined effect created constraints could incorporated genetic algorithm via penalty function approach instance nurses preferred working days thus partial solutions many day higher fitness however combining leads shortages night therefore infeasible solutions situation mall problem similar yet complex two types constraints dealt size constraints number constraints aickelin dowsland presented simple unsuccessful pyramidal genetic algorithm problem pyramidal approach best described hierarchical distributed genetic algorithm cascading clusters built bottom optimising larger parts problem thus hierarchy within one string rather subpopulations optimise different hence search larger search space lower resolution whilst search smaller search space higher resolution applied nursescheduling problem following way agents fitness based cover requests grade respectively agents fitness based cover requests grades agents optimise cover requests agents solve original problem full structure illustrated figure strings lower populations cascaded upwards using suitable crossover selection mechanisms instance fixed crossover points used agent combined one forms full solution although full problem epistatic less interaction nurse grades partially ignored compatibility problems combining parts reduced pyramidal structure hierarchical gradual combining using approach improved solution quality comparison standard genetic algorithm recorded however quality solutions still short produced tabu search far roulette wheel selection based fitness rank used choose parents fitness calculated using substitute fitness measure based requests cover detailed possibility qualified nurses covering ones partially ignored unsatisfied constraints included via penalty function paper investigate various partnering strategies agents improve upon results population pastes together form full solution figure nurse problem pyramidal structure similar nurse problem solution mall problem represented string many elements locations mall element indicates located mall geographically split different regions instance north east south west central objectives regional size shop synergy effects attractiveness area whereas others global total number shops certain type size application pyramidal structure mall problem follows along similar lines nurse problem line splitting string partitions nurses grade string split areas mall thus shops one area combined create larger parts mall finally full solutions however question arises calculate substitute fitness measure partial strings solution chosen pseudo measure based area dependant components global aspects taken account substitute fitness partial string calculated thus measure rent revenue created parts mall taking account constraints area based constraints ignored penalty function used account unsatisfied constraints due complexity fitness calculations limited overall population size refrained using several levels hierarchical design nurse scheduling instead simpler two level hierarchy used shown figure five optimising five areas separately one main population optimising original problem special crossover selects one solution figure mall problem pyramidal structure remainder paper investigate ways try improve previously found poor results suggesting ways combining partial strings intelligently alternative particularly mall problem would gradual without increasing overall population size would lead hence smaller however gradual approach might enabled algorithm find good feasible solutions slowly joining together promising building blocks contrast relatively harsh design building blocks succeed immediately exploring exact benefits gradual subsolutions would make another challenging area possible future research partnering strategies problem pick crossover partners noted competitive coevolutionary algorithms many strategies presented literature summarised instance bull paper seven strategies compared effectiveness fighting epistasis pyramidal genetic algorithm optimising nurse scheduling mall problems method used far algorithms agents assigned score based closely possible contribution partial string full solutions agents ranked within selection follows roulette wheel scheme based ranks aickelin dowsland random agents choose mating partners randomly amongst paired bull fogarty best strategy agent paired currently best agent case tie agent lower population index chosen potter jong distributed idea behind approach match agents similar ones paired previously ackely littman achieve subpopulation spaced evenly across single toroidal grid subsequently agents paired others grid location appropriate subpopulations children created inserted adjacent grid location said beneficial search process consistent pressure emerges since offspring appear parents neighbourhoods husbands algorithms use local mating neighbourhood set eight agents surrounding chosen location joined nature species carry others internally relationship propagated generation generation iba thus agent represents complete solution parts joined together case results solving original problem traditional parallel genetic algorithm means subpopulations use full fitness function evaluation selection attractiveness five strategies described far general make use problem specific knowledge however growing body research stanley wolpert macready well previous work suggests approaches exploit problem specific knowledge achieve better results pairing done strategy however pair accepted probability proportional fitness substitute fitness combined probabilities scaled substitute fitness fcomb equal greater fitness fbest pairing automatically accepted otherwise probability fcomb fbest mall problem inverse nurse scheduling partner choice approach exploits problem specific knowledge inspired idea presented ronald solves royal roads multiobjective optimisation problems using genetic algorithm first parent chosen following standard rules proportional fitness however second parent chosen according fitness depending attractiveness first parent measured different scale approach slightly different first parent still chosen according rank rather picking one agent appropriate subpopulation second parent ten candidates chosen random second parent chosen one creates fittest children first parent experimental results model allow fair comparison parameters strategies used problems kept similar possible total population agents split size main population size nurse scheduling respectively size mall problem principle two types crossover take place within parameterised uniform crossover genes coming one parent takes place crossover used appropriate parts assembled instance nurse problem agents would parent new child children created via uniform remainder crossover fixed point cases choice exists nurse problem new agents created four different ways either situations equal probability child created either way parent selection followed seven strategies outlined new solution created undergoes mutation bit mutation probability mutation would reinitialise bit feasible range algorithm run generational mode accommodate structure better every generation worst parents replaced fitness function calculations fitness score described used constraint violations penalised dynamic penalty parameter adjusts depending sub difference best best feasible agent population full details type weight calculated found smith tate aickelin dowsland stopping criterion top showing improvement generations obtain statistically sound results experiments conducted runs problem instances experiments started set random seeds initial populations results presented feasibility cost respectively rent format feasibility denotes probability finding feasible solution averaged problem instances cost rent refer objective function value best feasible solution problem instance averaged number instances least one feasible solution found algorithm fail find single feasible solution runs one problem instance censored observation one hundred nurse case zero mall problem made instead minimising cost nurses maximising rent mall equivalent poor solution problem cost represents sum unfulfilled nurses requests unfavourable shiftpatterns worked mall values rent thousands pounds per year results table shows results found algorithms two problems nurse problem mall problem using seven different partnering strategies combination pyramidal structure results compared found standard genetic algorithm sga aickelin dowsland tabu search results dowsland nurse problem theoretical bounds mall problem referred bound number interesting observations made nurse scheduling case sga approach failed find good even feasible solutions many data sets explained high degree epistasis present inability unmodified genetic algorithm deal pyramidal structure selection introduced results improve significantly however still found tabu search mall problem situation different results found sga fairly good high feasibility indicates higher number feasible solutions problem solution quality seems reasonably good however addition pyramidal structure results marked deterioration results different results explained nurse scheduling objective function value partial solution obtained summing cost values nurses involved furthermore able define relatively meaningful scores exploiting cumulative nature covering constraints due grade structure hence substitute fitness scores calculated allowed effective recombination partial solutions nursescheduling problem thus good correlation agent hence rank chance selected likelihood form part good solution also explains random scheme produces worse results best strategy although giving better results random selection fails solve many problems however closer observation experiments showed solved single data sets well indicates genetic variety important fitness evolution good solutions distributed joint strategies fail provide better solutions selection distributed strategy similar random strategy ignores fitness scores selection choosing fixed pool benefits results better complete random choice joint strategy works almost well shows principle dividing conquering works well nurse problem split along grade boundaries slightly poorer results explained full evaluation although parts passed thus correlation described lost two best strategies outperforming partner selection based attractiveness choice confirms partial scores good criterion selection pyramidal algorithm overall better corresponds higher selection pressure turn higher selection pressure conclude seems problem good correlation agents pyramidal structure good full solutions exist hence scheme highest selection pressure using problem specific information scores best however results also show even scheme compete tabu search results something discuss concluding section mall problem situation complicated since unlike nurse problem large part objective function source epistasis proposed partitioning string eliminate fully constraints second source epistasis contrast objective function depend largely whole string instance total number shops particular size allowed adding shops sizes areas known solution feasible unsurprisingly combination partial solutions often unsuccessful usually violates overall constraints solutions extremely unlikely feasible overall problem covered one fifth string equally unlikely solutions main population formed five feasible although solutions high rent ignore main constraints combination unlikely produce overall feasible solution situation slightly better solutions formed agent agent main population usually even agent main population feasible children even though partial string agent high rent usually incompatible rest string resulting many shops types thus contrast nursescheduling problem scores far poorer predictor compatibility parts form complete solutions confirmed average performance random strategy extremely poor results found best strategy similarly distributed strategy performs well giving credit idea even selection pressure without relying fitness scores whereas joint strategy performs poorly suffering unsuitable scores hindering pyramidal structure overall real winners complex strategies choice attraction first seems contradictory rely heavily upon subfitness scores however apart initial selection first parent subsequent fitness calculations made combining agents since mall pyramid two layers combinations always full solution hence full fitness score used thus direct link high fitness good solutions two performs better seems show certain amount randomness still important might indication lower predictive quality subfitness scores cost feas rent feas bound sga table experimental results nurse mall conclusions paper shown effect different partner strategies pyramidal genetic algorithm solving two different optimisation problems area scheduling result five simple strategies differ problems reflection accurateness measure sense predictive power subsolutions form full solutions following pyramidal recombination strategies therefore case nurse problem good match usefulness recombination simple strategies worked well whereas mall problem poorer correlation two problems distributed partnering strategy gives consistent results leads question whether form fitness sharing might beneficial partnering strategies taking idea ongoing research pyramidal genetic algorithms two advanced strategies use problem specific knowledge work well problems worked well nurse problem scores meaningful also worked well mall problem partners chosen based fitness score recombination case equals full original fitness score thus choosing parents evaluating possible children overcome possible shortcomings measure noted overall none partnering schemes managed outperform tabu search algorithm come close bounds hoped however one remember tabu search uses highly routines also relied heavily certain criteria present actual data used different random data would probably longer hold therefore without adding specific component would possible reach level solution quality conclusion pyramidal genetic algorithms without benefit greatly right choice partnering strategy improves solution quality negates possible shortcoming chosen scores references ackley littman altruism evolution communication brooks maes eds artificial life mit press aickelin genetic algorithms multiplechoice optimisation phd dissertation university wales swansea united kingdom aickelin dowsland exploiting problem structure genetic algorithm approach nurse rostering journal scheduling aickelin dowsland indirect genetic algorithm approach nurse scheduling review journal computing operational research bean noon ryan salton selecting tenants shopping mall interfaces bull evolutionary computing environments partners proceedings seventh international conference genetic algorithms morgan kaufmann fogarty coevolving communicating classifier systems tracking albrecht reeves steele eds artificial neural networks genetic algorithms new york davidor epistasis variance viewpoint foundations genetic algorithms rawlins morgan kaufmann jong genetic algorithms function optimisers whitley editor foundations genetic algorithms san mateo morgan kaufmann publishers deb genetic algorithms function optimisation genetic algorithms soft computing dowsland nurse scheduling tabu search strategic oscillation european journal operational research fogel evolutionary computation fossil record ieee press fuller tackling scheduling problems using integer programming master thesis university wales swansea united kingdom holland adaptation natural artificial systems ann arbor university michigan press husbands distributed coevolutionary genetic algorithms optimisation fogarty evolutionary computing iba emergent multiple agents using genetic programming voigt ebeling rechenberg schwefel eds parallel problem solving nature ppsn springer berlin johnson private communication martello toth knapsack problems wiley chichester potter jong coevolutionary approach function optimisation davidor schwefel manner eds parallel problem solving nature ppsn iii springerverlag berlin ronald selection meets seduction eshelman proceedings international conference genetic algorithms morgan kaufmann publishers san francisco smith tate genetic optimisation using penalty function proceedings icga forrest morgan kaufmann stanley ashlock testatsion iterated prisoner dilemma choice refusal partners langton artificial life iii redwood city wolpert macready free lunch theorem search santa institute santa
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dec examples local cohomology modules ramified regular local rings finite set associated primes rajsekhar bhattacharyya abstract lyubeznik conjecture remark asserts finiteness set ssociated primes local cohomology modules regular rings case ramified regular local ring open recently theorem proved noetherian regular local ring fixed ideal associated primes local cohomology hji finite contain paper use result construct examples local cohomology modules ramified regular local ring finitely many associated primes introduction consider noetherian ring ideal integer let hii local cohomology module support ideal fourth problem asked whether local cohomology modules noetherian rings finitely many associated prime ideals several important examples finiteness associated primes local cohomology modules regular rings prime characteristic regular local affine rings characteristic zero unramified regular local rings mixed characteristic smooth algebra zpz bblsz results support lyubeznik conjecture remark conjecture let regular ring ideal local cohomology module hii finitely many associated prime ideals recently shown excellent regular ring dimension containing field characteristic zero ideal ass finite lyubeznik conjecture open ramified regular local ring recently significant progress see fact theorem proved noetherian regular local ring fixed ideal associated primes local cohomology hji finite contain paper use result construct examples local cohomology modules ramified regular local ring finitely many associated primes section review basic results need proving main results section prove main results first proposition prove new variant result theorem using proposition theorem prove first main result consider ramified regular local ring mixed characteristic dimension choose regular system mathematics subject classification key words phrases local cohomology parameters ideal generated let ideal generated monomials formed regular system parameters consider ideal polynomial rings indeterminates prime field generated similar monomials formed indeterminates perfect ass hji finite every except important note monomials generated regular system parameters obviously system parameter contain one parameter consequence theorem corollary next important result stated follows consider ramified regular local ring dimension let positive integers consider set regular system parameters ideal generated ideal generated size minors matrix formed regular system parameters every except ass hji finite theorem present third main result construct examples ramified regular local rings countable set ideals countable set local cohomology modules supports ideals finite set associated primes basic results section discuss two basic results need prove results next section local cohomology modules following well known result regarding behaviour associated primes local cohomology modules faithfully flat extension lemma let faithfully flat local cohomology modules arbitrary every ideal ass finite set ass finite set next state following known result see theorem need prove proposition next section theorem let regular commutative noetherian local ring mixed characteristic set associated primes hji contain finite every every ideal main results begin section following proposition new variant result theorem prove next theorem part proof proposition similar theorem proposition let ramified regular local ring mixed characteristic let ideal suppose exists unramified regular local ring following properties eisenstein extension eisenstein polynomial ideal multiplication hii isomorphism ass ass hji ass hji finite proof element short exact sequence get following long exact sequence local cohomologies hii hii hii since non zero divisor hii long exact sequence get hii hii moreover since multiplication hii isomorphism hii hii unramified regular local ring eisenstein polynomial form every first claim sequence see let implies thus contradiction thus sequence using discussion see also sequences set consider following diagram short exact sequences every row column exact diagram yields following diagram long exact sequences rows columns exact hii hii hii hii hii hii hii hii hii hii hii first paragraph proof already hii thus diagram surjective let ker hii hii exists image map hii since surjective come hii via map hii due injectivity multiplication map image hii zero since every square diagram commutative get nonzero divisor hii hir using theorem conclude state first main result theorem let ramified regular local ring dimension mixed characteristic consider set regular system parameters ideal generated let ideal generated monomials formed regular system parameters assume polynomial ring indeterminates prime field characteristic monomials similar formed indeterminates generates perfect ideal every except ass hji finite proof since completion faithfully flat extension using lemma sufficient prove theorem complete ramified regular local ring assume complete fix regular system parameters let ideal generated number regular system parameters let coefficient ring hypothesis get system parameters turns unramified regular local ring algebraically independent think power series ring variables see part proof iii theorem theorem set exists eisenstein polynomial actually minimal polynomial see proof theorem set first would like show proceed follows since exists maximal ideal since get gives local moreover since regular local ring dim dim similar way dim dim every minimal primes inside otherwise one maximal ideal gives last equation becomes dim dim hand dim dim since domain also get dim dim dim dim combining results get dim dim set consider ideal generated monomials similar generates since element short exact sequence get following long exact sequence local cohomologies noted natural map consider polynomial ring variables prime field let ideal generated monomials similar generates assumption perfect set clearly faithfully flat extension easy see due flatness perfect ideal proposition get proposition section iii get every except gives multiplication isomorphism every except next observe following faithfully flat extension noetherian rings let ideal grade grade let maximal length sequence inside easy observe grade grade let element sequence let implies gives moreover thus maximal length aregular sequence inside contradiction thus grade grade faithfully flat extension tensoring last long exact sequence homology get moreover using results paragraph get grade grade since ring height grade ideal coincides thus get every except moreover gives multiplication isomorphism every except paragraph proof multiplication isomorphism every except moreover long exact quence get nonzero divisor every except thus proposition get every except ass hji finite finish proof need show grade grade argue following way local exists one maximal ideal contains every primes contains true thus grade grade similar true pass local ring let grade grade assume exist two sets sequences maximal elements ideals respectively change regular sequence may inside passing ring find get two sets sequences claim maximal prove claim let element let gives turn implies contradicts fact maximal sequence see maximal sequence let exists sequence first observe zero otherwise becomes part regular sequence assume gives thus passing ring get thus thus forms sequence replacing get forms sequence contradicts maximality sequence thus sets sequences maximal thus second paragraph proof notice thus every except ass hji finite thus conclude observe next important result application theorem following corollary corollary consider ramified regular local ring dimension mixed characteristic let positive integers consider set regular system parameters ideal generated ideal generated size minors matrix formed regular system parameters every except ass hji finite proof consider polynomial ring xij xij brevity field matrix indeterminates let xij ideal generated size minor matrix xij get xij perfect result immediate theorem remark theorem noted minimal polynomial ring even inside radical let raising power integer get btn contradiction example give examples local cohomology modules nonzero divisor generally multiplication local cohomology modules isomorphism section examples local cohomology modules nonzero divisor proof theorem already seen certain examples ideals multiplication local cohomology modules support ideals isomorphism present another source examples upon multiplication local cohomology modules isomorphism consider ideal since element short exact sequence get following long exact sequence local cohomologies hji hji hji since expansion zero ideal every hji thus long exact sequence reduces hji hji every present third main result paper following theorem proves always exists ramified regular local ring countable collection ideals get countable collection local cohomology modules support ideals finite set associated primes theorem let countable collection ideals unramified regular local ring mixed characteristic every non zero divisor every every countable subset exists ramified regular local ring mixed characteristic homomorphic image expanded ideal corresponding ass hji finite every true every proof first observe always include countable collection ideals stated hypothesis along corresponding let unramified regular local ring mixed characteristic since element short exact sequence get following long exact sequence local cohomologies since non zero divisor every every long exact sequence get assume complete regular local ring know every ideal every set ass countable since regular true ass module set ass ass ass ass countable get exists element belong element construction sequence set thus also becomes sequence consider sequence let ideal since regular sequence get also sequence fact get ass ass thus second paragraph proof get well sequence set since find thus part regular system parameters regular local ring moreover otherwise implies since regular sequence thus contradiction shows multiple ramified regular local ring mixed characteristic moreover also nonzero divisor thus using discussion find well also sequence consider following diagram short exact sequences every row column exact diagram yields following diagram long exact sequences every rows columns exact diagram surjective since regular let ker exists image map since surjective come via map due injectivity multiplication map image zero since every square diagram commutative get nonzero divisor ramified regular local ring set using theorem get ass hji finite references bblsz bhatt blickle lyubeznik singh zhang local cohomology modules smooth finitely many associated primes inventiones mathematicae arxiv bhatacharyya note associated primes bockstein homomorphisms local cohomologies ramified regular local rings communicated burch codimension analytic spread proc cambridge philos soc hochster solid closure commutative algebra syzygies multiplicities birational algebra contemp vol hochster eagon rings invariant theory generic perfection determinantal loci amer math huneke sharp bass numbers local cohomology modules transaction american mathematical society vol huneke problems local cohomology free resolutions commutative algebra algebraic geometry sundance utah res notes math jones bartlett boston lyubeznik finiteness properties local cohomology modules application dmodules commutative algebra invent math lyubeznik applications local cohomology characteristic reine angew math matsumara commutative ring theory cambridge university press certain rings differentiable type finiteness properties local cohomology journal algebra volume april pages arxiv peskine szpiro dimensionprojective finie cohomology locale puthenpurakal associated primes local cohomology modules regular rings arxiv dinabandhu andrews college garia kolkata india address rbhattacharyya
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jan stable homology associative rings olgur celikbas lars winther christensen liang greg piepmeyer abstract analyze stable homology associative rings obtain results artin algebras commutative noetherian rings study develops similarly classes simplicity discuss latter stable homology broad generalization tate homology vanishing stable homology detects classes gorenstein rings original domain tate homology closely related gorensteinness rings auslander modules show vanishing stable homology detects modules finite first characterization modules terms vanishing homology alone stable homology like absolute homology tor theory two variables computed flat resolution one module together injective resolution betrays stable homology balanced way tor balanced fact prove ring gorenstein stable homology balanced introduction homology theory studied paper introduced vogel vogel publish work theory appeared print paper goichot called homology name suggests theory generalization tate homology modules finite group rings vogel goichot also considered generalization tate cohomology studied detail avramov veliche paper theory called stable cohomology emphasize relation stabilization module categories align terminology henceforth refer homology theory stable homology modules ring stable homology family fit exact sequence abelian groups tor tor tori tori groups torr tori form respectively unbounded homology standard absolute homology thus led study stable unbounded homology simultaneously investigation takes cues studies stable cohomology absolute homology results look quite different comes inherent asymmetry definition date december mathematics subject classification key words phrases stable homology tate homology gorenstein ring research partly supported simons foundation collaboration grant mathematicians award nsa grant nsfc grant part work done stay texas tech university support china scholarship council thanks department mathematics statistics texas tech kind hospitality celikbas christensen liang piepmeyer stable homology present either precursors manifests different ways apparent results ring acts right consider stable homology tor left paper left distinguish right speak modules opposite ring study stable homology associative rings obtain conclusive results artin algebras commutative noetherian rings following overview denotes artin algebra commutative noetherian local ring results discuss special cases results obtained within paper internal references given parentheses one expects homology theory detect finiteness homological dimensions first two results reflect asymmetry definition stable homology right vanishing finitely generated following conditions equivalent finite projective dimension tor iii tor left vanishing finitely generated following conditions equivalent finite injective dimension tor iii tor two vanishing results reveal stable homology balanced way absolute homology tor balanced balancedness following conditions equivalent finite injective dimension finitely generated finitely generated isomorphisms tor tor iii isomorphisms tor tor another way phrase part say topic commutative noetherian ring gorenstein regular macaulay local ring regular every prime ideal follows commutative noetherian ring finite dimension gorenstein gorenstein ring need finite dimension consider example nagata regular ring infinite krull dimension appn exa came surprise balancedness stable homology detects gorensteinness outside local situation commutative noetherian ring isomorphic finitely tor gorenstein tor generated turns one several ways stable homology captures global properties rings indeed stable homology associative rings vanishing stable homology detects commutative noetherian necessarily local ring regular gorenstein finitely generated modules noetherian ring auslander bridger introduced homological invariant called noetherian ring integer every finitely generated left right module see avramov martsinkovsky sec tate homology defined modules rings shown agree stable homology iacob generalized tate homology setting includes case finitely generated finite prove tate homology let finitely generated finite isomorphic dimension every stable homology functor tor tate homology functor tor two primary generalizations modules finitely generated gorenstein projective dimension gorenstein flat dimension defined terms resolutions gorenstein projective flat modules notions introduced enochs jenda collaborators see holm general known gorenstein projective modules gorenstein flat finite iacob definition tate homology tor gorenstein projective dimension show tate homology agrees every gorenstein stable homology tor projective gorenstein flat example illustrate widely various homology theories differ example let commutative artinian local ring assume gorenstein denotes field example ring let injective hull residue field absolute homology torr every stable homology tori zero every unstable homology torr agrees tori every defined tate homology tor since gorenstein infinite see thm explains see also explains first syzygy space whence vanishing torr would imply finite projective dimension hence finite see thm left vanishing see yields exact sequence gives finally return topic vanishing much studied question gorenstein homological algebra detect finiteness finitely generated module commutative noetherian ring zero satisfies extia extia homa canonical map homa homa isomorphism original definition last requirement dispensed shown jorgensen enochs jenda consolidated work chaps deal almost exclusively gorenstein rings holm gorenstein homological algebra associative rings standard reference topic basic definitions recalled celikbas christensen liang piepmeyer characterizations modules finite literature none expressed solely terms vanishing homology functors hence excited discover let commutative noetherian ring finitely generg holds ated finite tor tensor products complexes rings paper assumed associative algebras commutative ring always possible works commutative artin algebra one take artinian ring finitely generated recall situation functor homk injective hull yields duality categories finitely generated finitely generated denote category work complexes index homologically follow standard notation see appendix covered complex differential integer symbol denotes homology degree denotes cokernel ker complex called acyclic morphism complexes induces isomorphism called quasiisomorphism symbol used decorate also used isomorphisms derived categories tensor product degree term differential given contrast standard tensor product complex construction first appeared similar constructions hom also treated great detail avramov veliche recall hom constructions appendix also study interactions one definition consider graded degree component endowed degree homomorphism defined elementary tensors becomes complex called unbounded tensor product contains tensor product subcomplex quotient complex called stable tensor product denoted definition yields exact sequence notice bounded complexes bounded side unbounded tensor product coincides usual zero tensor product stable tensor product stable homology associative rings note unbounded tensor product product totalization double complex collect basic results unbounded stable tensor products unbounded tensor product proofs mimic proofs tensor product one obtains results stable tensor product via isomorphisms additive right exact functors preserve split exact sequences functors preserve homotopy functors morphism yields isomorphisms cone cone cone cone follows cone acyclic similarly lemma let double let cycle product totalization assume contains component satisfies every every boundary product totalization proof goal prove existence sequence holds one product totalization assume positive constructed holds cycle product totalization one thus horizontal cycle hence horizontal boundary exists therefore element induction argument yields elements symmetric argument using yields proposition let let bounded acyclic acyclic bounded acyclic acyclic celikbas christensen liang piepmeyer proof product totalization double complex let upper bound let integer let cycle degree component cycle hence boundary assumed acyclic moreover assumption acyclic every ensures condition lemma met condition satisfied due boundedness thus boundary arbitrary follows acyclic similar argument roles exchanged handles case homology consider homology unbounded stable tensor product complexes notation differs slightly one employed goichot definition let let projective resolution let injective resolution torr unbounded homology modules stable homology modules tor two projective resolutions similarly two injective resolutions homotopy equivalent follows definitions independent choices resolutions notice finite projective dimension finite injective dimension every one torr tori hence tor standard liftings module homomorphisms morphisms resolutions imply definitions torr tori functorial either argument every functors torr tori functors homological sense every short exact sequence modules every give rise connected exact sequence stable homology modules tor tor tor tor analogous sequence tor modules similarly every short exact sequence every give rise connected exact sequence tor tori tori tori analogous sequence tor modules details see sec stable unbounded homology entwine absolute homology stable homology associative rings every every exact sequence complexes yields exact sequence homology modules tor tor torr torr flat resolutions like absolute homology unbounded stable homology computed using flat resolutions place projective resolutions proposition let flat resolution injective resolution isomorphisms torr tor every natural either argument proof let projective resolution every complex cone acyclic cor particular cone acyclic follows proposition acyclic view thus cone acyclic cone morphisms desired isomorphisms follow standard verify natural view next result immediate corollary every finite flat dimension tor holds every remark every flat finite projective dimension corollary covered wide classes rings flat modules finite projective dimension rings see thm generally rings finite finitistic projective dimension see jensen prop different direction rings cardinality see gruson jensen thm hand recall product fields von neumann regular ring every module ring flat osofsky shows sufficiently large product fields infinite global dimension hence must flat modules infinite projective dimension next result analogue stable homology thm proposition let let following conditions equivalent torr isomorphism connecting morphism tor every torr every injective every iii torr every proof every injective every one tor thus implies following use notation soft truncation iii let injective resolution let projective resolution let denote mapping cone let injective assumption one torr celikbas christensen liang piepmeyer right exactness tensor product acyclic hence acyclic proposition thus every one torr notice one iii immediate exact sequence torr vanishing detects finite flat dimension next result immediate consequence proposition parlance says flat copure flat dimensions agree tor corollary let tor sup torr injective dimension shifting useful tool computations stable homology module denote mth syzygy projective resolution mth cosyzygy injective resolution schanuel lemma uniquely determined summand view exact sequences one gets tor tor tor moreover flat resolution isomorphisms tor tor follows corollary suppose commutative flat every projective resolution flat resolution module thus follows proposition stable homology tor functor similarly injective resolution also injective resolution whence tor functor vanishing stable homology tor open partial converse corollary recall example prop finitely presented module flat projective theorem let finitely generated projective resolution following conditions equivalent finite tor iii tor homk faithfully injective tor moreover left noetherian idr finite equivalent iii tor stable homology associative rings proof implications iii iii clear see part follows iii dimension shifting use theorem get tor homk homk zero hence finite projective dimension prop thus ext finally assume left noetherian idr finite let faithfully injective view already proved sufficient show implies tor homk follows tor assumptions every free finite injective dimension thus every projective finite injective dimension yields exact sequence follows tor homk homk isomorphisms tor tor homk implies tor vanishing tor corollary let artin algebra duality functor finitely generated following conditions equivalent finite finitely generated tor iii tor proof part follows view part iii follows dimension shifting finitely generated injective resolution finally one homk see first paragraph section follows iii theorem injective hull faithfully injective result bass murthy lem commutative noetherian ring regular every finitely generated module finite projective dimension corollary commutative noetherian ring following equivalent regular finitely generated tor finitely generated iii tor proof implications iii clear see iii every finitely generated finite projective dimension theorem regular finitely generated one tor tori dimension shifting follows regular tate flat resolutions show extra assumptions one without resolving second argument compute tor let choose injective resolution projective resolution composite complex cone acyclic homotopy equivalence independent choice resolutions construction functorial celikbas christensen liang piepmeyer lemma let bounded complex flat let projective resolution injective resolution derived category isomorphism functorial second argument proof short exact sequences forms commutative diagram whose rows triangles top triangle rotated back isomorphism induced composite one completion morphism triangles yields desired isomorphism via triangulated five lemma see prop functoriality second argument straightforward verify lemma let projective resolution injective resolution let complex flat acyclic every injective isomorphism functorial second argument proof assumptions proposition complex acyclic application exact sequence yields let denote mapping cone complex acyclic proposition application mapping cone sequence yields isomorphism desired isomorphism follows module functoriality second argument straightforward verify remark lemmas hold placed flat resolution objects discussed next paragraph appear literature variety names stick terminology acyclic complex flat called totally acyclic acyclic every injective tate flat resolution pair flat resolution totally acyclic complex flat tate name invoked resolutions used compute tate homology see thm left coherent ring tate flat resolution finite gorenstein flat dimension see prop noetherian every finitely generated finite gdimension tate flat resolution direct argument thm exists pair projective resolution stable homology associative rings acyclic complex finitely generated projective following properties complex acyclic every injective one homr homr last isomorphism holds finitely generated see prop thus totally acyclic complex flat tate flat resolution next result shows stable homology computed via tate flat resolutions apply section compare stable homology tate homology used proofs theorems theorem let tate flat resolution every every isomorphism tor functorial second argument proof choose isomorphic consider split exact sequence let projective resolution injective resolution set exact sequence acyclic follows proposition acyclic whence surjective morphism since totally acyclic complex flat modules lemma isomorphism moreover lemma yields isomorphism thus one explains second isomorphism next computation tor tor first isomorphism holds first equality follows proposition canonical surjection flat resolution discussed introduction homological invariant finitely generated modules noetherian rings characterizations modules finite traditionally involved vanishing homology invertibility certain morphism see example recently avramov iyengar lipman showed finitely generated module commutative noetherian ring finite isomorphic complex rhomr rhomr celikbas christensen liang piepmeyer derive category crucial step proof next theorem show implies isomorphism vanishing stable homology tor theorem let commutative noetherian ring finitely generated following conditions equivalent every finite flat dimension tor iii tor proof implication iii trivial module tate flat resolution theog rem yields isomorphisms tor follows induction flat dimension acyclic lem iii let finitely generated projective lution let injective resolution assumption complex acyclic explains third homr homr homr last isomorphism holds proposition homr equals homr follows thm thm finite corollary commutative noetherian ring following equivalent gorenstein finitely generated tor iii tori finitely generated proof result goto gorenstein every finitely generated finite cor combining theorem one gets equivalence implication iii clear finally follows iii dimension shifting balancedness stable homology absolute homology balanced ring always isomorphisms torr torr follows already corollary stable homology balanced special rings indeed artin algebra stable homology balanced dual module finite projective dimension whence converse part corollary open section technical lemma similar result enochs estrada iacob thm recall notation lemma let acyclic complexes flat flat rmodules respectively integers isomorphisms stable homology associative rings proof complexes acyclic hence prop yields demonstrates first isomorphism statement second proved similarly finally two isomorphisms connect exact sequences one obtains last isomorphism via triangulated five lemma see prop theorem let tate flat resolutions isomorphism tor tori proof let tate flat resolutions respectively choose isomorphisms every one tor tor first last isomorphisms follow theorem second fourth isomorphisms follow dimension shifting third isomorphism holds last isomorphism lemma definition let stable homology balanced one tor tor theorem says stable homology balanced pairs rmodules tate flat resolutions every module every tate flat resolution see thm stable homology balanced pairs corollary artin algebra following conditions equivalent stable homology balanced iii stable homology balanced finitely generated celikbas christensen liang piepmeyer proof per part implies clearly implies iii let dual module injective finitely generated thus stable homology balanced finitely generated modules follows corollary pdr well finite duality idr finite corollary commutative noetherian ring gorenstein stable homology balanced finitely generated proof gorenstein ring every finitely generated module finite cor every finitely generated module tate flat resolution see thus balancedness stable homology follows theorem cong every finitely versely balancedness stable homology implies tor generated gorenstein corollary corollary commutative noetherian ring finite krull dimension gorenstein stable homology balanced proof gorenstein finite krull dimension therefore stable homology balanced per converse holds corollary vanishing stable homology tor artin algebra duality vanishing stable homology tor understood via vanishing tor proposition let artin algebra duality functor finitely generated following conditions equivalent idr finite tor finitely generated iii integer tor tor proof implications iii clear see part follows iii dimension shifting finitely generated finally vanishing tor implies corollary finite whence idr finite follows commulocal rings analyze vanishing stable homology tor tative noetherian rings start locally let commutative noetherian local ring residue field depth invariant defined depthr inf extir finitely generated injective dimension computed idr sup extir stable homology associative rings depth finite ring exists finitely generated module finite injective dimension consequence new intersection theorem due peskine szpiro roberts following analogue thm lemma let commutative noetherian local ring residue field every finitely generated every isomorphism torr homk extjr extr particular torr space proof let finitely generated projective resolution let injective resolutions definition torr ith homology complex compute using proposition follows homr homr homr next simplify using adjointness homr homr homr homr homr homr homr homr homk homr homr finally pass homology may remark lemma suggests stable homology tor vector space indeed ring annihilates two homotopic multiplication projective resolution case injective resolution case see definition hence multiplication zero unbounded stable homology particular stable space homology tor proposition let commutative noetherian local ring residue field let finitely generated space finite rank finite injective dimension gorenstein tor proof space torr finite rank finite rank assumption exact sequence finitely generated vector spaces extjr depthr idr see roberts thm torr finite rank follows lemma finite injective dimension compared characterization globally gorenstein rings corollary condition iii sharper theorem let commutative noetherian local ring residue field following conditions equivalent gorenstein tor celikbas christensen liang piepmeyer finite rank iii tor finite exists finitely generated tor holds rank tor proof implications iii clear see let module specified theorem yields result holm thm gorenstein idr also finite follows proposition gorenstein remark let commutative noetherian local ring residue field let denote injective hull faithfully injective computation based theorem shows ranks tor simultaneously finite ext homr rankk tor rankk tor rankk homr ext rankk ext combined equality ranks thm yield characterizations regular complete intersection gorenstein local rings terms example regular size stable homology spaces tor holds equivalently gorenstein tor finite rank equivalently tor commutative rings commutative prime ideal local ring flat follows stable homology modules tor lemma let commutative noetherian ring let let prime ideal let every natural isomorphism tor tor implies tor prime ideals hence tor proof let injective resolution let projective resolution flat resolution second isomorphism follows proposition computation gives similarly one gets desired isonow five lemma yield morphisms follow injective resolution matlis theory proof next result similar compared lemma noetherian hypothesis dropped used invoke matlis theory stable homology associative rings lemma let commutative ring let let prime ideal let every natural isomorphism tor tori implies tor prime ideals hence tor theorem let commutative noetherian ring let finitely holds idrp generated tor finite every prime ideal follows proof hypotheses lemma one tor proposition local ring gorenstein idrp finite implies however gorenstein vanishing tor pdrp finite corollary theorem idrp finite next corollary immediate per remarks corollary commutative noetherian ring finitely generated tor corollary let commutative noetherian ring finite krull dimension finitely generated following conditions equivalent idr finite tor iii tor proof implications iii clear see part follows iii idr equals sup idrp prime ideal dim remark know assumption finite krull dimension implies corollary necessary theorem vanishing tor locally finite injective dimension imply finite injective dimension consider gorenstein ring infinite krull dimension vanishes ring hand know tor comparison tate homology section compare stable homology tate homology parallel findings avramov veliche first recall definitions acyclic complex projective called totally acyclic acyclic every projective complete projective resolution diagram totally acyclic complex projective projective resolution isomorphism see sec let complete projective resolution tate homology see iacob sec tor celikbas christensen liang piepmeyer complete projective resolution finite gorenstein projective dimension see thm noetherian finitely generated complete projective resolution finite complete projective resolution finitely generated surjective see thm lemma let complete projective resolution following conditions equivalent acyclic every injective isomorphisms functors tor tor proof assume acyclic every injective pair tate flat resolution see follows theorem isomorphic tor functors tor converse let injective one tor tori hence theorem let noetherian let finitely generated complete projective resolution isomorphisms functors tor tor proof module complete projective resolution finitely generated see isomorphisms show complex acyclic every injective lemma finishes argument remark isomorphisms homology modules theorem actually follow one isomorphism unapparent proof rests theorem finitely generated module complete projective resolution finitely generated surjective see thus kernel ker bounded complex projective modules given module let cone one would construct proposition complex acyclic lemma proof yield acyclic lemma gives combining isomorphisms gives desired one without extra assumptions ring know stable homology agrees tate homology whenever latter defined general relation stable homology tate homology tied unresolved problem gorenstein homological algebra theorem explains called gorenstein projective exists totally acyclic complex projective see similarly module called gorenstein flat exists totally acyclic complex flat see stable homology associative rings theorem following conditions equivalent every gorenstein projective gorenstein flat every complete projective resolution isomorphisms functors tor tor proof assume every gorenstein projective gorenstein flat let complete projective resolution follows totally acyclic complex flat modules see emmanouil thm thus stable homology tate homology coincide lemma converse let gorenstein projective let totally acyclic complex projective since isomorphisms functors tori tori follows acyclic every injective thus totally acyclic complex flat gorenstein flat remark holm notes prop obvious way achieve every gorenstein projective gorenstein flat ensure pontryagin dual every injective flat every flat finite projective dimension first condition satisfied left coherent second discussed remark description rings gorenstein projective modules gorenstein flat seems elusive see sec complete homology thesis triulzi considers homological functor torr torr construction similar mislin covariant ext nucinkis contravariant ext resulting homology theory called complete homology like stable homology generalization tate homology compare two generalizations point view stable homology interesting know agrees complete homology latter universal property direction main results two homology theories agree rings finitely generated modules artin algebras complete commutative local rings moreover two theories agree tate homology whenever defined exact condition every gorenstein projective module gorenstein flat see theorem appendix start recalling definition stable cohomology let following denote homr subcomplex homr degree term homr homr called bounded hom complex quotient complex homr hom called stable hom complex projective resolutions extir homr celikbas christensen liang piepmeyer called bounded cohomology stable cohomology modules hom ext stable cohomolavramov veliche use notation ext ogy notation standard tate cohomology coincides stable cohomology whenever former defined see cor proposition let bounded homr acyclic complex homr acyclic bounded homr acyclic complex homr acyclic proof similar proof proposition standard isomorphisms study composites functors hom extent possible establish analogs standard isomorphisms composites hom see sec seems analog adjunction prop setup propositions namely let complex let complex let complex finiteness conditions unbounded tensor product associative proposition complexes either following conditions complexes finitely presented modules bounded complex isomorphism functorial proof every one modules canonical homomorphism bounded homomorphisms isomorphisms inner products finite recall thm functor commutes products finitely presented thus canonical homomorphisms isomorphisms complexes finitely presented modules straightforward verify isomorphisms commute differentials form isomorphism complexes stable homology associative rings model following swap isomorphism proof similar proof proposition uses functor homr commutes coproducts finitely generated proposition complexes either following conditions complexes finitely generated modules bounded complex isomorphism homs homs functorial next result analog prop used establish duality stable homology stable cohomology proposition let assume complex finitely presented morphism homs homs functorial furthermore isomorphism complex projective modules complex injective modules proof every one compute follows homs homs homs homs isomorphism holds module finitely presented every see thm hand every one homs homs homs homs homs set homs homs homomorphism given straightforward verify morphism functorial celikbas christensen liang piepmeyer finally module projective module injective follows prop invertible morphism invertible lemma let bounded complex finitely generated projective let complex bounded every injective isomorphism derived category homk homk hom functorial proof set homk commutative square horizontal morphism dual canonical embedding vertical map isomorphism proposition morphism standard evaluation map square induces morphism triangles homotopy category cone cone construction functorial three arguments recall say natural cone cone coker cone coker hom yield desired isomorphism derived category homk homk hom regard functoriality notice morphism arguments say induces solid commutative square following diagram coker coker rrr rrr cone cone cone cone coker coker stable homology associative rings commutativity derived category dashed square consequence functoriality arguments handles similarly lemma immediately yields useful duality theorem let assume degreewise finitely generated projective resolution every injective every isomorphism homk homk ext tor functorial prove next two results one proceeds proof proposition proposition let complex finitely generated let morphism homr homr functorial furthermore isomorphism following conditions complex finitely generated projective modules complex finitely presented modules complex flat modules complex projective modules proposition let complex let complex let complex finitely presented morphism homr homr functorial furthermore isomorphism complex projective modules pinched tensor products christensen jorgensen devised pinched tensor product compute tate homology view theorem proof thm applies verbatim yield next result refer reader definition pinched tensor product theorem let tate flat resolution let acyclic complex set every isomorphism tor next corollary analogue corollary corollary let commutative let gorenstein flat rmodules corresponding totally acyclic complexes flat modules holds acyclic respectively tor complex flat following statements equivalent totally acyclic complex flat holds every injective tor celikbas christensen liang piepmeyer conditions hold gorenstein flat corresponding totally acyclic complex flat proof follows definition pinched tensor products complex flat tori holds complex acyclic theorem totally acyclic acyclic every injective holds every injective tor finally follows definition pinched tensor products isomorphism references maurice auslander mark bridger stable module theory memoirs american mathematical society american mathematical society providence luchezar avramov foxby homological dimensions unbounded complexes pure appl algebra luchezar avramov srikanth iyengar joseph lipman reflexivity rigidity complexes commutative rings algebra 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cambridge university press cambridge university missouri columbia current address university connecticut storrs address texas tech university lubbock address url http lchriste lanzhou jiaotong university lanzhou china address lliangnju columbia basin college pasco address pggreg
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journal machine learning research submitted published inferring learning neuronal correspondences ashish kapoor akapoor microsoft research one microsoft way redmond usa jan paxon frady efrady department neurobiology university california san diego usa stefanie jegelka stefje department electrical engineering computer science massachusetts institute technology cambridge usa william kristan kristan department neurobiology university california san diego usa eric horvitz horvitz microsoft research one microsoft way redmond usa editor tbd abstract introduce study methods inferring learning correspondences among neurons approach enables alignment data distinct multiunit studies nervous systems show methods inferring correspondences combine data effectively studies make joint inferences behavioral decision making possible data single animal focus data collection machine learning prediction representative invertebrate nervous system european medicinal leech acknowledging computational intractability general problem identifying correspondences among neurons introduce efficient computational procedures matching neurons across animals methods include techniques adjust missing cells additional cells different data sets may reflect biological experimental variation methods highlight value harnessing inference learning new kinds computational microscopes multiunit neurobiological studies keywords pca neurobiology metric learning correspondence matching probabilistic ashish kapoor paxon frady stefanie jegelka william kristan eric horvitz kapoor frady jegelka kristand horvitz introduction neurobiologists long pursued understanding emergent phenomena nervous systems neuronal basis choice behavior much research neuronal systems grapples complex dynamics interactions among multiple neurons new techniques calcium imaging dye vsd imaging cacciatore gonzalez tsien electrode recordings enable larger views nervous systems however many experimental preparations amount data collected via tedious experiments limited example data voltagesensitive dyes bleaching dyes also neuronal damage caused phototoxicity developed methods combining data multiple experiments pool data neural function approach allows make inferences data sets impossible obtain individual preparations coalescing data multiple experiments intrinsically difficult problem difficulty matching cells roles across animals variation observed nervous systems individual animals based developmental differences well artifacts introduced preparation execution experiments developing means identifying correspondences cells across animals would allow data pooled multiple animals supporting deeper inferences neuronal circuits behaviors focus specifically experimental studies neurons composing ganglia verbana briggman leech stereotypical nervous system consisting repeating packets neurons third neurons identified neurons found reliably different animals remaining twothirds neurons yet identified believed maintain similar properties functional roles across animals general problem correspondence matching cells two different animals illustrated figure seek identify neurons equivalent across ganglia obtained different animals example red cell animal several candidate correspondences animal varying degrees similarity indicated shades red ideally find match via jointly considering multiple similarities two animals problem mapped bipartite task solved optimally munkres however want jointly solve matching task larger numbers animals matching across multiple graphs defined individual nervous systems intractable problem class papadimitriou steiglitz addition problem even difficult matching must also take account variations numbers properties neurons observed different animals variations due developmental differences neurons may missing duplicated experimental artifacts neurons may plane focus destroyed delicate dissection key challenge endeavor formulation similarity measure takes account physical parameters cells size location well functional properties inferring learning neuronal correspondences figure challenge identifying correspondence among neurons ganglia different animals given compatibility constraints correspondence algorithm seeks mapping neurons two verbana preparations goal find correspondences cells animals color coding illustrates compatibility constraints feasible matches highlighted red green blue cells animal found animal indicated matched colors degree feasibility matches depicted via shading cells animal compatible cell source neuron animal highlighted white border although figure shows two animals compatibility constraints occur across pairs animals used kapoor frady jegelka kristand horvitz table proposed algorithmic framework step learn compatibility measure step recover correspondence map step infer missing data given training data pairs match cells estimate parameter matrix defines compatibility function ith neurons animal pairs start initialized empty match set iteratively determine next best match solving equation update end cells matched construct matrix aggregates data animals row corresponds cells permuted according matching use probabilistic pca infer missing data machine learning framework use set neuronal data collected briggman consists optical vsd recordings populations neurons segmental ganglion six different leeches earlier research data identified neurons involved decisionmaking particular study aimed understanding roles neuronal populations decisions swim crawl following stimulation sensory neurons nerve stimulated way would elicit equal likelihood swimming crawling previous study considered single cell activations joint analysis neurons using dimensionally reduction techniques pca lda however techniques limited one animal time current study propose framework analyzes data increase power analysis rather using handcrafted measure employed framework relies supervised training data algorithm estimates appropriate similarity function neurons different animals based training set highconfidence correspondences correspondences readily identified neurons nervous systems verbana muller important capability algorithm take account probabilistic nature inferred correspondences algorithm begins learning weighting function relevant features maximizes likelihood matches within training set next step approach jointly solve correspondence matching problem neurons across animals considering potential missing extra cells animal final step consider correspondences functions fill missing data demonstrate pooling neurophysiological data multiple studies principled manner leads larger effective data greater statistical power individual studies inferring learning neuronal correspondences specifically pipeline methodology includes three steps detailed table determining similarity score across pairs cells recovering correspondences consistent similarity measure estimating missing data describe steps detail learning similarity measure cells goal step learn similarity function indicating feasibility match ith cell animal cell animal desirable characteristic function high positive value likely matches diminishing values poor matches characteristic captured exponentiation negative distance measure among sets features represent multiple properties cells formally dimensional feature representations individual cells animal summarizes physical size location etc functional optical recordings properties parameter matrix positive entries learned data intuitively negative log similarity function distance function feature representations zero distance two feature vectors result highest similarity measure whereas representations distance away feature space diminishing value matrix parameterizes distance measure given training data consisting several probable pairs matched neurons use solve optimization problem describe details following list features used work structural features absolute position cell respect entire observed frame relative position cell relation entire ganglion absolute size cell pixels indicator vector specifying packet neuron located among central left anterior left posterior right anterior right posterior central posterior packet relative position coordinate neuron respective packet functional features coherence electrophysiological observations swim oscillations single cell discrimination time see briggman distinguishes swim versus crawl behavior intuitively optimization problem finds parameter minimizes distance pairs cells tagged matches maximizing distance among pairs formally parameter compatibility measure estimated minimizing objective arg min log log log subject constraint entries positive sum labeled training pairs tagged likely matches intuitively first term log objective prefers solutions would collapse distance matched pairs kapoor frady jegelka kristand horvitz zero rest terms prefer solution distance rest cells maximized optimization straightforward simple gradient descent always find locally optimal solution note appropriate constraint positive condition however suggest using constraint due simplicity optimization almost reduction performance pipeline correspondence matching second step calculate correspondence matches across animals instead calculating matches simultaneously framework follows iterative procedure future matches made using similarity function also comparing geometric structural relationship candidates past matches besides considering distances induced similarity function log unlike past work graph matching williams bunke proposed method utilizes knowledge landmarks inducing constraints impose topological geometric invariants algorithm considers iteration denotes set already determined matches algorithm determines next set neurons animals matched solving following optimization task arg min log pairs parameter balances compatibility measure landmark distances dlm matches recovered prior iterations landmark distance computation provides important structural topological constraints solving correspondence tasks given anchor points landmark distances attempt capture structural locational relationship respect available landmarks several options commute distance mckay lovasz graph euclidean distance computed considering either locations feature representation neurons experiments compute landmark distances neuron animal neuron animal respect set anchor points dlm log log optimization problem equation solved using energy minimization procedures boykov minka set newly discovered matches included process repeated matches stay settle essentially goal find set matched neurons across animals objective function minimized start reasonable initialization solution example solving consecutive pairs animals solution iteratively refined considering data drawn one study time searching replacement neuron would lower total energy replacements continue minimization observed utilizing landmarks appropriate informative signal matching neurons leech typified geometric structure although soma positions vary animal animal often certain somas remain arranged particular geometrical inferring learning neuronal correspondences figure cell correspondences inferred across six verbana graphics show results correspondence matching procedure across six animals color coding indicates correspondences matched cells across different animals share color highlight two cells depicted show matches lines linking neurons across animals several cells remain unmatched depicted using dashed lines unfilled interior algorithm capable handling partial matches cells present six animals due true structural differences losses either preparation sensing relationships instance nut cells typically form pattern sensory neurons usually arranged along packet edge often wrap around cell types arrangements useful identification cells eye extend algorithm utilize relationships framework extended handle poor matches missing cells considering sink cell every animal sink cell fixed cost matching denoted acts threshold neuron matches costs greater disallowed sink cells soft representation probability particular neuron visible given preparation pooling across animals finally third step framework reconstructs data corresponding cells missing remain unobserved animals particular consider kapoor frady jegelka kristand horvitz figure computed canonical ganglion verbana derived correspondence matching algorithm used results correspondence matching algorithm generate average canonical ganglion computing mean location size cell matched across least three different animals shades neurons colored according weight determined lda projection would distinguish swim crawl models brighter color mean higher weight colors used arbitrary trophysiological activity unobserved cells latent random variables infer latent variables exploiting fact observed animals correspondence information across animals fill missing electrophysiological data formally invoke data completion via probabilistic principle component analysis ppca roweis tipping bishop apply ppca construct matrix row corresponds neuron column corresponds fluorescence intensity short time interval since correspondences animals calculated stack data animals rows arranged according discovered correspondences use denote absence data due missing cells animal ppca algorithm recovers low dimensional structure data inserts missing data via expectation maximization inferring learning neuronal correspondences dempster ppca algorithm starts initialized projection alternates missing data estimated considering statistical relationships data estimates projection refined consider matrix dimensions consists neuronal activity recordings cells animals constructed using methodology described text first scale values matrix lets denote lowdimensional representation data matrix dimensions principal components dimensions ppca algorithm first initializes matrices randomly alternates following two steps estimate refine xnew cnew xnew xnew xnew algorithm converges maximum change individual dimensions estimates less ppca guaranteed converge produces data completion even neurons observed animals implementation optimization step see table performed via limited memory bfgs liu nocedal routine energy minimization equation performed via iterative variational inference beal three parameters need specified framework upper limit cost allowed matches parameter compatibility relative locality measure dimensionality low dimensional projection ppca parameters determined via methodology cross validation performed considering aggregated matrix randomly reducing observed data considering reconstruction error using norm removed data process repeated times parameters minimum average reconstruction error chosen search space parameters lie try linear range experiments training data learning parameter collected experimentalist epf different match pairs across animals fig shows resulting compatibility measure data note physical properties size relative location packet membership likely matches highlighted white outline illustrate quality learned function matching procedure results correspondence map fig matching neurons across different animals correspondence map calculated used generate prototypic model animal averaging physical well functional properties fig resulting correspondence map computed simultaneously across animals provides simple way analyze quality recovered solution addition physical properties functional characteristics two matched neurons across different animals similar across animals figure observed simple estimator kapoor frady jegelka kristand horvitz figure activity neurons leech ganglion prior study briggman showing neuronal activity used identify homologous neurons across animals dye traces two different neurons considered matches correspondence matching algorithm traces highlight algorithm capability recovering correspondence across cell functionally similar based average activity neurons five animals predicted activity sixth one figure estimated entire time series activity given cell animal considering activity corresponding cell across rest five animals two different models swim crawl mode computed prediction performed computing average across observed six animals lower differences observed electrophysiological activity predictions made model learned rest animals confirm framework recovered correct correspondences neurons across animals although matching algorithm performs quite well likely algorithm far perfect many matched cells may correct since functional responses cells factor matching cells little functional signal harder match big signals cells lacking functional signal however providing lot information predicting behavioral outcome thus cells likely poor matches also likely cells decision circuit also possible many cells effectively given data set matches truly reflect homologous pairs expect many cells functionally mismatching similar cells may hurt analysis correspondence matching algorithm enables pooling data across animals allows exploration feasible previously example figure shows inferring learning neuronal correspondences figure bar graphs highlight results test recovered correspondence using analysis plots generated first considering candidate test animal building predictive model cell animal swam crawled using remaining five animals compares error predicted observed electrophysiological activity matching using proposed framework random selection differences across six test cases significant projection recovered applying isomap tenenbaum dimensionality reduction method extension linear methods pca algorithm applied entire pooled data recovered dimensions consistent across animals thus visualized analyzed within reference frames previously application techniques pca lda briggman limited single animal time resulting dimensions incomparable across animals pooling data enabled methodology proved valuable predictive models decision making figure shows pooling data across animals enable earlier predictions one two behaviors swimming crawling following stimulation data single animal specifically pca performed pooled data earliest discrimination time swim crawl determined according procedure described briggman figure highlight cells composed canonical ganglion play important role behavioral decisions animal combining data across multiple animals enables transfer overlay information allowing aggregation important statistical parameters robust empirical models figure shows ganglion maps six animals highlighting cells contribute towards discrimination amongst swim crawl trials note highly discriminative cells towards red spectrum consistent physical properties location size across different animals also note cells significantly different cell identified earlier studies briggman kapoor frady jegelka kristand horvitz figure using correspondences predict behavior neuronal activity identification corresponding neurons across animals enables larger data sets constructed pooling observations multiple preparations turn enable deeper accurate data analysis address questions interest figure shows projections generated applying isomap algorithm blue red dots correspond swim crawl mode depict trajectory dye trajectories take animal note isomap applied individual animal might result projections inconsistent across different animals however using discovered correspondences neurons across animals combine data six animals recover projections consistent animals related work work described paper builds upon many different machine learning particular key ingredients include metric learning correspondence matching probabilistic dimensionality reduction roweis tipping bishop distance metric learning fairly active research area work distance metric learning focus neighbor classification scenario duda often aim learn mahalanobis metric consistent training data frome weinberger davis distance metric learning method employed paper closest work goldberger globerson roweis modified consider sets similar cells given user inferring learning neuronal correspondences figure bar graphs showing pooled data allows discriminate swim crawl significantly earlier reported earlier using pca analysis data single animal briggman correspondence problems employed multitude applications computer vision particularly closer scenario among simplest transformations rigid bodies geometry exploited goodrich mitchell mcauley correspondences among objects objects pose significant challenges algorithms applied general correspondence problems largely combine compatibility points features local geometric compatibility matches models formulated graphical models mcauley torresani starck hilton selecting nodes association graph cho cour extended criteria duchenne zass shashua lee methods consider laplacian constructed neighborhood graph umeyama escolano mateus models learned full training examples torresani closest idea using reference points approaches based seed points sharma landmarks jegelka strategies starck hilton guessing points help orient remaining points rigid body mcauley caetano kapoor frady jegelka kristand horvitz conclusion future work proposed methodology likely even useful combination analyses example model learned past data employed guide future experimentation computing correspondences model data ongoing experiment use model guide information extraction strategies methodology also extended perform withinleech analysis discovering bilateral pairs neurons addition methodology readily used analyze simultaneous activity multiple neurons animals foresee valuable uses approach overlaying data larger nervous systems moving beyond cells abstractions nervous system organization components retina columns vertebrate nervous systems given simplicity appeal potentially pooling large quantities data correspondence methodology may find wide use many areas neuroscience acknowledgments acknowledge assistance johnson apacible erick chastain paul koch references hertz shental weinshall learning mahalanobis metric equivalence constraints journal machine learning research beal variational algorithms approximate bayesian inference university college london boykov veksler zabih fast approximate energy minimization via graph cuts ieee transactions pattern analysis machine intelligence briggman abarbanel kristan optical imaging neuronal populations science bunke recent developments graph matching cacciatore brodfuehrer gonzalez jiang adams tsien kristan kleinfeld identification neural circuits imaging coherent electrical activity dyes neuron cho lee lee reweighted random walks graph matching european conference computer vision cour srinivasan shi balanced graph matching advances neural information processing systems davis kulis jain sra dhillon metric learning international conference machine learning inferring learning neuronal correspondences dempster laird rubin incomplete data via algorithm journal royal statistical society series duchenne bach kweon ponce algorithm graph matching computer vision pattern recognition duda hart stork pattern classification john wiley sons escolano hancock lozano graph matching entropic manifold alignment computer vision pattern recognition frome singer malik image retrieval classification using local distance functions advances neural information processing systems globerson roweis metric learning collapsing classes advances neural information processing systems goldberger roweis hinton salakhutdinov neighbourhood components analysis advances neural information processing systems gonzalez tsien voltage sensing fluorescence resonance energy transfer single cells biophysical journal goodrich mitchell approximate geometric pattern matching rigid motions ieee transactions pattern analysis machine intelligence jegelka kapoor horvitz interactive approach solving correspondence problems international journal computer vision lee cho lee matching via reweighted random walks computer vision pattern recognition liu nocedal limited memory method large scale optimization mathematical programming lovasz random walks graphs survey combinatorics paul erdos eighty mateus horaud knossow cuzzolin boyer articulated shape matching using laplacian eigenfunctions unsupervised point registration computer vision pattern recognition mcauley caetano fast matching large point sets occlusion pattern recognition mcauley caetano barbosa graph rigidity cyclic belief propagation point pattern matching ieee transactions pattern analysis machine intelligence kapoor frady jegelka kristand horvitz mckay practical graph isomorphism congressus numerantium minka divergence measures message passing microsoft research technical report munkres algorithms assignment transportation problems journal society industrial applied mathematics papadimitriou steiglitz combinatorial optimization algorithms complexity prentice hall englewood cliffs roweis algorithms pca spca sharma horaud cech boyer shape matching based diffusion geometry seed growing computer vision pattern recognition starck hilton correspondence labelling surface matching international conference computer vision tenenbaum silva langford global geometric framework nonlinear dimensionality reduction science tipping bishop journal royal statistical society torresani kolmogorov rother feature correspondence via graph matching models global optimization european conference computer vision umeyama eigendecomposition approach weighted graph matching problems ieee transactions pattern analysis machine intelligence weinberger blitzer saul distance metric learning large margin nearest neighbor classification advances neural information processing systems williams wilson hancock multiple graph matching bayesian inference pattern recognition letters zass shashua probabilistic graph hypergraph matching computer vision pattern recognition inferring learning neuronal correspondences figure determining influential cells using linear discriminant analysis ganglion maps experiments shown maps experiments fig cells based magnitude contribution linear discriminant direction red yellow represent large magnitude contributions blue represents small contributions see include cell marked using white least cells influential arrow
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strongly verbally closed groups andrey mazhuga jul faculty mechanics mathematics moscow state university moscow leninskie gory msu abstract recently proven free many virtually free verbally closed subgroups algebraically closed group establish sufficient conditions group extension free group group satisfying law algebraically closed group verbally closed apply conditions prove fundamental groups closed surfaces except klein bottle almost free products groups satisfying law algebraically closed group verbally closed introduction subgroup group called verbally closed see also equation form solution solution subgroup group called algebraically closed system equations form solution solution subgroup group called retract semidirect product normal subgroup easy see retract algebraically closed subgroup algebraically closed subgroup verbally closed subgroup thus question naturally arises conditions subgroup group revers implications hold known see reverse implications hold general following established finitely presented finitely generated algebraically closed subgroup retract finitely generated equationally algebraically closed subgroup retract work author supported russian foundation basic research project group finitely generated finite subset group equationally noetherian system equations coefficients finitely many unknowns equal finite subsystem class verbally closed subgroups similar structural descriptions known however generated free groups see generated free nilpotent groups see situation rather simple verbally closed subgroups algebraically closed subgroups retracts things article following theorem established theorem let group let verbally closed virtually free infinite subgroup containing infinite abelian noncyclic subgroups algebraically closed finitely generated retract call strongly verbally closed group algebraically closed subgroup group containing verbally closed subgroup notice assertion given theorem describes certain class strongly verbally closed groups particular free groups belong class abelian groups another class strongly verbally closed groups see corollary article establish conditions group extension free nonabelian group group satisfying strongly verbally closed conditions used establish strong verbal closedness rather wide class groups instance section apply prove following fundamental groups closed surfaces except klein bottle strongly verbally closed groups satisfying law strongly verbally closed notice fundamental group closed surface linear admits faithful representation see linear therefore equationally noetherian thus theorem fundamental group closed surface klein bottle retract finitely generated group containing verbally closed subgroup proceeding formulation main result introduce notation symbol means cyclic group generated symbols mean verbal subgroups generated word respectively symbol means normal closure symbols mean center derived subgroup respectively main theorem group exists short exact sequence groups free group group satisfying law divisible subgroup infinite group means free product two groups order two let element free group basis say group satisfies law let element free group basis verbal subgroup group determined generation set elements system equations unique solution strongly verbally closed hence use theorem one choose one discretion generating set law central divisible subgroup shown case direct factor however neither need use fact set words note sets may elements want draw reader attention fact strongly verbally closed group verbal closedness group given assumption consider question groups given group verbally closed restrict following remark follows example free group strongly verbally closed hand observation group generating set consisting elements verbally closed free subgroup proof let verbally closed free subgroup rank let generating set consisting elements let decomposition elements natural number relatively prime orders ordg elements decomposition consider equation xpm equation solution easy see pmi mod ordg solution equation solution since element proper power thus contradicts verbal closedness free subgroup subgroup verbally closed since free factor free factor retract therefore transitivity verbal closedness verbally closed get contradiction section discuss examples corollaries showing main theorem applied section contains proof main theorem argument sophisticated version proof main theorem also based use lee words let notations elements group xky denote respectively derived subgroup center group denoted respectively commutator two elements group subset group hxi mean cardinality subgroup generated centralizer respectively notations hxin mean cyclic group order cyclic group generated element respectively index subgroup group denoted subgroups group symbol means normal closure subgroup group symbols mean free product groups free product groups amalgamated subgroup semidirect product groups direct product groups respectively free group rank basis sometimes instead notation use abridged notation rewriting example examples corollaries following example particular theoretical interest included demonstrate main theorem applied concrete group example group free group freely generated strongly verbally closed proof consider following short exact sequence groups choose generating set law group next choose words verify conditions main theorem hold consider following words show system unique solution straightforward verify solutions equation either form clear solution second equation substituting whence since assume word reduced last letter letter last equality means thus solutions two equations system form substituting third equation obtain equality follows similar shows one choose words words following two examples special cases assertion stated theorem example free group strongly verbally closed proof let free group abelian strongly verbally closed accordance corollary let free group consider following short exact sequence groups choose conditions main theorem trivial example virtually free cyclic group unique extraction roots strongly verbally closed proof virtually free cyclic group clear exists short exact sequens groups form free group group let set elements following established virtually free group virtually cyclic element decomposes product two elements follows immediately statement generating set uniqueness solution equation assumed hypothesis example remaining conditions main theorem trivial corollary let short exact sequence groups free group group satisfying law divisible group exist elements strongly verbally closed proof let put verify conditions main theorem need check system unique solution clear system solution let solution system since thus therefore need following technical lemma lemma let short exact sequence groups free group group satisfying law free subgroup proof group free subgroup free group subgroup normal even fully invariant verbal subgroup since word therefore abelian easily deduce theorem free group normal cyclic subgroup thus free group elements corollary let free product groups satisfying law strongly verbally closed proof consider following short exact sequence groups kernel natural epimorphism cartesian subgroup cartesian subgroup free group clear free let commutator let law assume indices words contain letters easy see case law shall verify conditions corollary group free lemma therefore exist elements cartesian subgroup trivially intersects free factors therefore since subgroup cartesian subgroup elements belong subgroup form hig center group therefore group decompose proper free product subgroup theorem see would imply subgroup hig thus sice fundamental group connected surface free see example strongly verbally closed see fundamental group closed surface decompose proper free product nevertheless corollary fundamental group closed surface except klein bottle strongly verbally closed text article surface means closed connected compact without boundary surface fundamental group orientable surface genus denote surface type fundamental group surface genus denote surface type following technical lemma rather lemma let fundamental group surface whose euler elements commute free group euler characteristic euler characteristic proof proof see example see free group therefore following epimorphisms naturally since commutator subgroups images epimorphisms groups follows commutator subgroups also proof clear groups abelian therefore corollary groups strongly verbally closed group excluded hypothesis corollary thus remains consider surfaces whose euler characteristic negative consider following short exact sequence groups assertion lemma free group assuming verify conditions corollary lemma free subgroup therefore elements suppose exists element say assertion lemma elements therefore since elements free group equality means commute contradicts choice whence proof main theorem following lemma well known lemma subgroup group finite system equations form solution solution algebraically closed proof denote new letters interpret variables example solvability equation equivalent solvability system xyz recall integer matrix reduced diagonal matrix integer elementary transformations means system equations form reduced system form means sequence transformations form follows form transformations system solution group containing subgroup system solution therefore following lemma holds lemma subgroup group finite system form solution solution algebraically closed corollary abelian group strongly verbally closed proof let verbally closed abelian subgroup group suppose system form solution since verbally closed subgroup equation system solution let tuple solution equation since abelian thus easy see solution recall lee word variables free group rank element free group rank obtained simultaneous conjugation exists vis elements generate cyclic subgroup words constructed integers actually easy see lee result implies existence universal lee word variables lemma positive integer exists element properties hold free groups even proof assertion follows immediately lee result following simple fact embeds malnormal subgroup subgroup fact follows result free group set satisfying condition freely generates malnormal subgroup thus lee word universal suitable also proof following lemma based use universal lee words lemma let verbal subgroups generated word respectively free group verbally closed subgroup group finite system equations form solution solution proof let system form solution hypothesis lemma sides equations decompositions form decomposition side equation let set elements present normal form element set union sets elements since free group exist elements add elements set since word elements assume indices tuples contain letters element add system equation occurrence constant decompositions sides equations system replace word denote new system note sides equations contain constants sides constants system solution system solution let denote equations system concatenation tuples let universal lee word variables consider equation equation solution construction take following solution thus since verbally closed subgroup equation solution say construction word either form form clear values value element free group since free group indeed construction property lee words implies according property lee words exists solution system recall system obtained system carry inverse transformation set equalities precisely using equalities form sides equalities form replace according decompositions sides tuples call tuples form concatenation equations elements elements leads following set equalities whence solution system required ready prove main theorem proof main theorem let verbally closed subgroup group lemma show system form solution solution equation system associate system new unknowns consider system consisting equations systems clear system solution system solution using elements eui nui condition main theorem equation system associate system eui eui nui next consider system consisting equations systems clear system solution system solution since free group lemma eui eui clear system form therefore lemma system solution say turn accordance condition main theorem implies following equalities hold therefore elements following set equalities whence since central exist elements following equalities hold means solution elements accordance equality either form form suppose since divisible group rimi replace element tuple element leave elements unchanged since cental subgroup equalities remain unchanged suppose equation solution indeed equation solution since system verbally closed subgroup means since thus condition main theorem therefore thus tuple solution system required acknowledgments author thanks klyachko many useful conversations many useful remarks references baumslag myasnikov remeslennikov algebraic geometry groups algebraic sets ideal theory algebra campbell quick robertson smith groups andrews volume cambridge university press jaco certain subgroups fundamental group closed surface mathematical proceedings cambridge philosophical society vol issue klyachko mazhuga verbally closed virtually free subgroups math appear see also arxiv lee certain words free groups algebra lyndon schupp combinatorial group theory new york mazhuga free decompositions verbally closed subgroups free products finite groups journal group theory appear see also myasnikov roman kov verbally closed subgroups free groups journal group theory see also robinson course theory groups springer roman kov equations groups groups complexity cryptology roman kov khisamiev verbally existentially closed subgroups free nilpotent groups algebra logic stillwell classical topology combinatorial group theory springer wise residual finiteness positive groups comment math helv
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improved algorithm augmenting paths metric space haitao wang department computer science utah state university logan usa abstract aug let path graph vertices embedded metric space consider problem adding new edge diameter resulting graph minimized previously icalp problem solved time paper based new observations different algorithmic techniques present log time algorithm keywords phrases diameter path graphs augmenting paths minimizing diameter metric space introduction let path graph vertices embedded metric space consider problem adding new edge diameter resulting graph minimized problem formally defined follows let graph edge length length path total length edges path two vertices use denote length shortest path diameter defined maxu let path graph vertices edge connecting let vertex set assume metric space distance two vertices specifically following properties hold triangle inequality particular edge length equal assume given two vertices distance obtained time goal find new edge connecting two vertices add diameter resulting graph minimized problem studied solved problem time paper present new algorithm runs log time algorithm based new observations structures optimal solution different algorithmic techniques following previous work refer problem diameteroptimally augmenting path problem doap short related work path euclidean space constant also gave time algorithm find solution problem doap euclidean plane carulfel gave linear time algorithm adding new edge minimize continuous diameter diameter defined respect points vertices general problem many variations also studied see references therein consider general graph edges haitao wang licensed creative commons license leibniz international proceedings informatics schloss dagstuhl informatik dagstuhl publishing germany augmenting paths lengths integer goal general problem compute set new edges add resulting graph minimum diameter problem variants even approximation results given general problem many variations see upper bounds lower bounds values diameters augmented graphs also investigated see since diameter important metric network performance measures cost two nodes network discussed problem augmenting graphs minimizing diameter variations many practical applications data networks telephone networks transportation networks scheduling problems etc application problem doap consider following scenario transportation networks suppose highway connects several cities order reduce transportation time want build new highway connecting two cities distance farthest two cities using highways minimized clearly problem instance doap approaches tackle problem first gave log time algorithm decision version problem given value determine whether possible add new edge diameter resulting graph implementing decision algorithm parallel fashion applying megiddo parametric search solved original problem doap time differentiation referred original problem doap optimization problem improvement previous work twofold first solve decision problem time algorithm based log time algorithm previous work however discovering new observations problem structure help data structure avoid certain expensive operations eventually achieve time complexity second comparing decision problem algorithm optimization problem completely different previous work let diameter resulting graph optimal solution instead using parametric search identify set candidate values search using algorithm decision problem however computational difficulties arise approach due set large computing certain values certain values computing takes time circumvent difficulties algorithm several steps step shrink significantly always remains importantly step obtain certain formation based next step reduce several steps size reduced remaining values computed log time point use decision algorithm find additional log time equipped linear time algorithm decision problem utilizing several algorithmic techniques searching techniques data structure eventually solve optimization problem log time rest paper organized follows section introduce notation observations section present algorithm decision problem optimization problem solved section wang figure illustrating resulting graph new edge added preliminaries section introduce notation observations two vertices use denote edge connecting metric space hence length use denote resulting graph adding edge essentially let denote diameter goal optimization problem doap find pair indices minimized let diameter optimal solution given value decision problem determine whether words determine whether exist pair yes say feasible value recall graph refers length shortest path two vertices consider pair indices define follows refer fig definition define largest shortest path length vertices define largest shortest path length vertices define largest shortest path length define shortest path length verified also shown following observation holds observation max due triangle inequality metric space following monotonicity properties hold observation augmenting paths pair let denote subpath hence length algorithms need compute next observation gives algorithm result also given present proof completeness paper lemma time preprocessing given pair compute time compute log time proof preprocessing compute prefix sum array done time finishes preprocessing consider pair note computed constant time easy see min hence computed constant time consider function unimodal function specifically changes first increases decreases hence computed log time binary search sequence computing also done log time similar way omit details computing although one may able time clear make log time even log time preprocessing seen later major difficulty solving problem doap efficiently refer difficulty main effort circumvent difficulty providing alternative efficient solutions pair use denote cycle consider notice shortest path also shortest path hence two paths one uses edge use denote length second path notation min according definition summarize discussion following observation observation pair min following simplify notation context clear use index refer vertex example refers refers decision problem section present time algorithm solving decision problem value goal determine whether feasible whether equivalently wang figure illustrating changes whether pair yes algorithm also find feasible edge observation holds determine whether feasible algorithm determine whether exists fixed consider functions light observation monotonically increasing monotonically decreasing see fig define four indices follows refer fig definition define largest index define similarly define smallest index otherwise let define similarly discussed feasible observation following lemma lemma see fig proof according observation implies definitions see fig three cases similar figure illustrating changes computing light lemma compute time follows discuss case first according lemma computed constant time pair compute time following simple algorithm first compute done computing incrementally first time compute augmenting paths figure illustrating path dotted curve using edge compute incrementally first time next compute way total time correctness based monotonicity property lemma compute using similar approach log time algorithm since computing takes log time lemma following lemma give another approach needs time lemma computed time proof discuss case since case analogous key idea pair instead computing exact value sufficient determine whether follows show time preprocessing determine whether time index pair let smallest index implies preprocessing compute index easily done time even log time binary search consider pair goal determine whether vacuously true crucial observation length path using new edge less equal see fig clearly path length computed constant time thus determine whether constant time therefore determine whether constant time pair result use similar algorithm computing compute time lemma thus follows due difficulty mentioned section clear whether possible compute log time recall feasible exists computed known following use indirect approach determine whether intersection four intervals empty every determining feasibility define goal determine whether empty wang figure illustrating graph consider since known determine intersection constant time intersection empty know following assume intersection empty let smallest index intersection easy observation note actually implies since obviously true since otherwise according definition gave approach determine whether log time log time preprocessing following new observations help range minima data structure show whether determined constant time time preprocessing define largest index observe consider corresponding goal determine whether since talking essentially considering graph recall cycle observation min vai vertex cycle note vai see fig following lemma lemma either proof suppose consider prove must hold definition holds since obtain note min hence must hold proves one direction lemma suppose true either prove consider pair indices prove sufficient show thus obviously holds following assume implies hence either recall min thus definition hence obtain augmenting paths otherwise thus hence vai vai consequently obtain proves direction lemma recall define smallest index observe note hai hai due preceding lemma following lemma lemma either hai holds hai proof suppose hai need prove anything following assume hai consider hai goal show holds indeed since lemma either since hai hence must proves one direction lemma suppose either hai holds hai goal show consider lemma sufficient show either hai since obtain definition hai otherwise holds hai hai still otherwise hai thus holds proves direction lemma let denote total length cycle vai following observation crucial immediately leads algorithm lemma observation either hai hai proof suppose hai hai note hence equivalent therefore holds hai hai lemma observation follows lemma time preprocessing given corresponding whether determined constant time proof preprocessing first compute done time due monotonicity property compute also done time due monotonicity property next compute array let build data structure wang range minima data structure built time given pair minimum value subarray returned constant time finishes preprocessing step takes time total consider corresponding goal determine whether done time follows observation either hai hai since hai computed preprocessing check whether hai true yes done assertion otherwise need determine whether hai holds end first compute hai constant time querying data structure hai note computed constant time therefore determine whether time proves lemma lemma decision problem solved time proof following theorem summarizes algorithm theorem given determine whether feasible time feasible feasible edge found time proof first preprocessing lemma time compute time also preprocessing lemma next following compute intersection constant time intersection empty done otherwise obtain smallest index intersection stop algorithm assertion feasible report feasible edge otherwise use lemma determine whether constant time yes stop algorithm assertion feasible report feasible edge otherwise proceed algorithm stop check stop algorithm assertion feasible clearly spend time thus total time algorithm optimization problem section present algorithm solves optimization problem log time making use algorithm decision problem given section refer decision algorithm sufficient compute use decision algorithm find optimal new edge additional time start easy observation must equal diameter pair observation equal pair define let according discussion note smallest feasible value compute entire set since let smallest feasible value hence min following first compute log time using decision algorithm searching techniques augmenting paths computing convenience begin computing define matrix define otherwise observation following lemma shows sorted matrix sense row sorted descending order left right column sorted descending order top bottom lemma proof consider two adjacent matrix elements row observation hence either case holds consider two adjacent matrix elements column observation obtain note essentially equal equal clearly hence either case note element vice versa since smallest feasible value also smallest feasible value construct explicitly rather given evaluate log time since computed log time lemma using searching techniques find calling decision algorithm log times evaluating elements total time calling decision algorithm log total time evaluating matrix elements also log hence compute log time computing done similarly log time although corresponding sorted matrices may defined slightly differently omit details however compute log time way due difficulty mentioned section note essentially reduces search space compute min thus min hence done computing otherwise must need compute compute use similar way computing instead use following approach point success approach relies information implied computing case assume hence let new edge added optimal solution also call optimal edge since following observation observation optimal edge proof assume contrary observation equal one without loss generality assume wang figure illustrating changes three indices shown since must smallest feasible value however contradicts min respect define similar way defined section respect except change specifically define largest index defined similarly define smallest index otherwise defined similarly note similar monotonicity properties lemma also hold recall optimal edge easy observation since strictly larger intersection empty let smallest index intersection note since following lemma shows actually optimal edge lemma optimal edge proof pair let max observation max first prove following claim see fig one hand consider definition since observation hence since otherwise would equal incurring contradiction hand consider observation observation hence since claim hypothesis max therefore obtain implies hence claim follows proceed prove lemma based claim sufficient show follows assume contrary according definition hence let since value since feasible value recall smallest feasible value thus obtain contradiction since therefore holds lemma thus follows lemma crucial immediately suggests following algorithm first compute indices done time using similar algorithms computing section augmenting paths fact even afford log time compute indices hence simplicity use similar algorithm computing section instead one lemma total time log next compute smallest index intersection let set index interval intersection empty lemma leads following observation observation smallest feasible value set proof lemma one edges optimal edge observation thus smallest feasible value obtain following stronger result although observation sufficient algorithm lemma proof pair let max observation max first prove following claim indeed assume contrary definition hence let note feasible value however contradicts smallest feasible value next prove lemma using claim clam thus feasible value lemma know therefore smallest value lemma thus follows observation essentially reduces search space values tempting first explicitly compute set find set however due difficulty able compute set log time alternatively use following approach compute finding set recall according observation min hence equal therefore observation exists equal let based discussion smallest feasible value let smallest feasible value let smallest feasible value hence min using technique searching following lemma computes log time lemma computed log time wang figure illustrating graph whose diameter proof define matrix define otherwise easy verify row sorted ascending order left right column sorted ascending order bottom top consequently using searching technique found calling decision algorithm log times evaluating elements clearly given evaluate constant time hence computed log time recall min case done computing following assume thus help information implied compute log time details given let denote largest index subpath length strictly smaller note definition similar defined section except change let denote smallest index let subset hence thus define following lemma gives way determine lemma proof since discussions observation diameter graph equal length shortest path two vertices min see fig since know fact since otherwise would set contradicting smallest feasible value simplicity discussion assume since otherwise keep updating find note eventually found reach since prove following claim one hand since obtain since diameter graph min obtain follows augmenting paths hand assume contrary since feasible value clearly set however contradicts smallest feasible value proves claim claim show prove lemma first show indeed since claim based definition holds see fig since obtain implies thus since remains prove indeed recall note claim fact implies thus hence note consider prove sufficient prove follows recall since diameter recall min definition know since hence must proves lemma thus follows light lemma case smallest feasible value note number values hence compute easily found additional log time using decision algorithm first sorting values binary search next lemma gives algorithm compute time help data structure lemma computed time proof consider easy see length cycle hence obtain following max max min define dmin discussions dmin therefore computing boils computing dmin following compute dmin time computed additional time first compute easily done time due monotonicity properties recall already computed compute time checking whether next compute array clearly array computed time build data structure data structure built time given pair minimum value subarray computed constant time wang finally compute dmin constant time querying data structure therefore compute dmin thus compute time summary compute log time case overall algorithm computing optimal solution summarized proof theorem theorem optimal solution optimization problem found log time proof first compute log time using decision algorithm searching techniques compute min second using compute indices done time compute smallest index intersection add set initially hence computed time return compute log time lemma proceed compute lemma find smallest feasible value set log time finally return min computes log time applying decision algorithm eventually find optimal edge additional time references alon decreasing diameter bounded degree graphs journal graph theory bender lca problem revisited proc latin american symposium theoretical informatics pages luciano guido proietti improved approximability nonapproximability results graph diameter decreasing problems theoretical computer science carufel grimm maheshwari smid minimizing continuous diameter augmenting paths cycles shortcuts proc scandinavian workshop algorithm theory swat pages demaine zadimoghaddam minimizing diameter network using shortcut edges proc scandinavian conference algorithm theory swat pages frati gaspers gudmundsson mathieson augmenting graphs minimize diameter algorithmica frederickson johnson generalized selection ranking sorted matrices siam journal computing frederickson johnson finding kth paths generating searching good data structures journal algorithms gao hare nastos parametric complexity graph diameter augmentation discrete applied mathematics gudmundsson knauer smid stehn fast algorithms augmenting paths proc international colloquium automata languages programming icalp pages augmenting paths harel tarjan fast algorithms finding nearest common ancestors siam journal computing ishii augmenting outerplanar graphs meet diameter requirements journal graph theory mccormick edge addition problems operations research letters megiddo applying parallel computation algorithms design serial algorithms journal acm schoone bodlaender van leeuwen diameter increase caused edge deletion journal graph theory
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distributed control asymptotic synchronization dynamical networks tao liu aming cao bclaudio persis bjulien hendrickx jan department electrical electronic engineering university hong kong hong kong china faculty science engineering university groningen groningen netherlands icteam institute catholique louvain belgium abstract paper studies synchronization dynamical networks communication firstly two estimators introduced node one estimate state estimate average state neighbours two estimators distributed rule etr dwell time designed network achieves synchronization asymptotically zeno behaviours designed etr depends information node obtain thus implemented decentralized way key words distributed control asymptotic synchronization dynamical networks introduction synchronization dynamical networks related systems attracted lot attention due extensive applications various fields see arenas ren details motivated fact connected nodes networks share information digital platform problems recently investigated circumstance nodes communicate neighbours certain instants use limited communication network resources effectively control etc see heemels reference therein introduced networked control systems extensively used synchronize networks circumstance node get limited information main issue becomes use limited information design etr node network achieves synchronization asymptotically material paper presented conference email addresses taoliu tao liu ming cao claudio persis julien hendrickx preprint submitted automatica meanwhile prevent zeno behaviours caused hybrid nature etc undesirable practice tabuada early works etc focused dynamical networks simple node dynamics dimarogonas johansson distributed etc used achieve consensus prevent zeno behaviour decentralized etr threshold introduced achieve consensus seyboth strategies proposed persis frasca shown robust skews local clocks delays limited precision communication recently attention increasingly paid networks generalized linear node dynamics different types etc developed achieve either bounded asymptotic synchronization networks demir lunze zhu liu meng chen xiao garcia yang order achieve asymptotic synchronization well prevent zeno behaviours two main methods developed literature one uses bidirectional communication event time node sends sampled state neighbours meanwhile asks neighbours current states update january consider dynamical network described control signal meng chen xiao uses unidirectional communication node sends sampled information neighbours require information neighbours liu garcia yang however latter needs estimators node uses exponential term ert oder prevent zeno behaviours hxi bui xin state node node dynamics matrix input matrix control input respectively generally continuous communication neighbouring nodes assumed aij yields following network paper study asymptotic synchronization networks generalized linear node dynamics using unidirectional communication method main differences existing results follows firstly new sampling mechanism used two estimators introduced node whereas existing results need every node estimate states neighbours secondly inspired method proposed tallapragada chopra replace exponential term extensively used literature dwell time originally introduced switched systems cao morse simplify implementation designed etr thirdly distributed etr node designed based two estimators dwell time whereas existing results use decentralized etrs consist local information node state error node estimator exponential term garcia yang introducing estimation synchronization errors neighbours using neighbours sampled information proposed etr method reduce number sampling times node significantly hxi aij paper assume connections realized via discrete communication node obtains information neighbours certain discretetime instants present version network study design etr node achieve asymptotic synchronization suppose topological structure network fixed undirected connected introduce two estimators ovi node used estimate state ovi used estimate average state neighbours adopt following control input lii control gain designed states ovi respectively state equations ovi given network model preliminaries notation denote set real numbers real numbers integers set real vectors real matrices identity matrix vector matrix entries respectively represents euclidean norm vectors also induced norm matrices superscript transpose vectors matrices kronecker product matrices single let undirected graph consisting node set link set link nodes say node neighbour node vice versa let aij adjacency matrix aii aij aji node node neighbours otherwise aij aji laplacian matrix lij defined lij lii aij ovi tki tki tki increasing time sequences tki represent time instants node sends updates neighbours receives updates one neighbours respectively assume time delay computation execution tki represents sampling time time node broadcasts updates communication network ideal circumstance time delays data dropouts therefore set tkj subset node receives updated information aij index set neighbours node vector represents deviation state estimator node easily compute network etr contains information available node node estimator therefore one estimator node insufficient implement etr practice overcome difficulty introduce another estimator node shown network theoretically equivalent network two estimators ovi etr equivalent etr implemented practice time sequence tki decided etr tki inf tki eventtriggering function designed tki node samples tki tki calculates tki sends tki neighbours reinitialize estimator tki tki addition node reinitialize tor ovi time receives updates neighbours assume network well initialized node samples sends neighbours therefore problem given network topology design proper etr network achieves synchronization asymptotically without zeno behaviours remark shown liu assumptions network estimators node number neighbours node also theoretically equivalent network thus equivalent network two estimators ovi hand error extensively used literature design etr node sends sampled state neighbours node sending tki instead tki turns reduce number estimators implementation new sampling mechanism needs information used literature instead calculating estimators method calculates node hence method implementation advantages particular networks large limited embedded computing resources node like existing results literature etc method node needs send neighbours rather relative state information extensively used network continuously interconnected nodes course important study network using relative state information design purposes studied future simplify analysis show network controller estimators equivalent following system node maintains estimator state neighbours hxi lij tki tki tki defining gives dynamics defined moreover paper use model etr analysis obtained results implemented using controller two estimators ovi etr based network give definition asymptotic synchronisation definition let rnn solution network initial condition network said achieve synchronization asymptotically every rnn following condition satisfied thus controller becomes lii lii lim kxi substituting gives network equivalent remark communication network ideal model ovi simplified complicated model needed describe network dynamics time delays packet loss influence synchronization performance however due robust property asymptotic synchronization bounded synchronization guaranteed moreover let lii etr reformulated tki inf tki shown trentelman necessary sufficient condition asymptotic synchronization network continuous interconnections existence positive definite matrices final synchronization error may depend time delay magnitude probability packet loss another important problem case conditions network still achieve synchronization asymptotically issues studied future control condition requires linear systems system matrices asymptotically stable simultaneously stronger stabilizable another alternative find common network etc regarded network external input disturbance according stability theory necessary condition system asymptotically stable corresponding system also described without term asymptotically stable hence existence matrix solutions lyapunov equations also fundamental requirement network etc achieve asymptotic synchronization paper assume matrices exist denote network rewritten since topology network undirected connected laplacian matrix irreducible symmetric one zero eigenvalue exists orthogonal matrix diag choose due zero row sum property defining gives let next give useful lemma used prove main result let diag diag defining gives lemma consider network following two inequalities hold kek kyk kyk kek proof due use properties supported facts denoting diag system simplified kzk kek kzk kek let diagonalizable simultaneously diag eigenvalues let eigenvalues matrix gives thus defining kxi last equality follows therefore asymptotically equal network achieves synchronization asymptotically result summarized following lemma combining gives inequality similar gives lemma system asymptotically stable network achieves synchronization asymptotically let based comparison theory khalil kel tkl whenever kel tkl tkl tkl tkl solution ordinary differential equation kpi bkk khk kbkk kbkk following result theorem network achieves synchronization asymptotically distributed etr tki inf tki kei initial condition tkl tkl kel tkl tkl setting tkl gives kel tkl tkl moreover zeno behaviour occurs network kel kel tkl tkl solving tkl shows take tkl positive time constant change values kel therefore requires least make kel move true kei hence kei suppose obtain contradiction always hold let infimum times hold due finite number nodes exists node kel times arbitrarily close kel follows etr must tkl tkl show exist tkl tkl establish since kei gives kei suppose obtain contradiction tkl case kel tkl continuity kel implies existence kel therefore holds kel contradiction infimum times kel select lyapunov function candidate diag derivative along system satisfies hand inequality gives kyk calculating kel kyk kyk kkek kek substituting gives tkl directly gives kel kkek kel khk kyk kyk kek kbkk bkk kyk therefore equilibrium point system asymptotically stable based lemma network achieves synchronization asymptotically use lemma get substituting gives kel kel kyk kyk inequality holds combining yields substituting yields combining gives leads proof etr existence tki tki guaranteed dwell time show asymptotic synchronization claim network satisfies kei remark like results synchronization dynamical network etc trentelman guinaldo one usually needs global parameters guarantee asymptotic synchronization exclude zeno behaviours parameters estimated using methods proposed related literature franceschelli initialized node beginning however use local parameters rather global ones replace using local parameter degree node remains open deserves attention example show effectiveness method consider network nodes parameters follows remark existing results demir lunze guinaldo seyboth zhu garcia yang adopt graph describe use decentralized etrs summathe topology rized following compact since matrix two eigenvalues imaginary axis complex plane network synchronize stable solution deter tki inf kei mined initial condition calculating get select figure gives simulation results network distributed etr ddt shows effectiveness prowhere three design parameters posed method figure give sampling obviously etr depends local information time instants first seconds clarity thefrom node asymptotic synchronization oretical value minimum maxbe achieved paper introduce imum sample periods node proposed etr term updated simulation time given table shows tkj estimates synchronization errors actual sample periods much larger neighbours continuously thus provides node useful information determining sampling times therefore proposed etr reduce sampling also compared method decentralized times significantly particular cases etr det proposed guinaldo large see example section details according remark bounded synchronization shown liu similar distributed guaranteed seyboth etr exponential term also case advantage method guarantee asymptotic synchronization however paclear compare method case per replaces exponential term dwell time asymptotic synchronization implemented easily practice gives also achieved select durupper bound designed etr therefore ing simulation period network modified etr also synchronize ddt samples times total whereas network network without zeno behaviours det samples times times total remark simplify notations paper considers case scalar however obtained results extend case directly pointed heemels joint design controller rule hard problem however select control gain synchronizes network stabilize simultaneously selected solving group linear matrix inequalities moreover periodic etc method proposed stabilize linear systems heemels triggering condition verified periodically paper check condition time interval tki tki check condition continuously period tki tki great interest study asymptotic synchronization using periodic etc communications fig simulation network ddt table sample period node node node node node node node node node demir lunze synchronziation systems ifac conf analysis design hybrid systems eindhoven netherlands dimarogonas johansson eventbased control systems ieee conference decison control shanghai china franceschelli gasparri giua seatzu decentralized estimation laplacian eigenvalues systems automatica garcia cao wang casbeer decentralized consensus linear multiagent systems directed graphs american control conference chicago usa guinaldo dimarogonas johansson sanchez dormido distributed control strategies interconnected linear systems iet control theory applications heemels donkers teel periodic control linear systems ieee trans automatic control heemels johansson tabuada introduction selftriggered control ieee conference decison control maui usa liu feng consensus linear systems distributed strategy ieee trans cybernetics khalil nonlinear systems edition prentice hall new jersey liu cao persis hendrickx distributed control synchronization dynamical networks estimators ifac workshop distributed estimation control networked systems koblenz germany meng chen event based agreement protocols networks automatica fax murray consensus cooperation networked systems proceedigs ieee ren beard atkins informaiton consensus multivechicle cooperative control collective behavior local interation ieee control system magazine seyboth dimarogonas johansson broadcasting average consensus automatica tabuada scheduling stabilizing control tasks ieee trans automatic control tallapragada chopra decentralized eventtriggering control nonlinear systems ieee trans automatic control trentelman takaba monshizadeh robust synchronization uncertain linear node conclusion paper studied asymptotic synchronization networks using distributed etc using introduced estimators distributed etr node explored relies state node states estimators shown proposed etc synchronizes network asymptotically zeno behaviours worth pointing data packet dropout common phenomena definitely affects synchronization networks communication appears synchronization networks imperfect communication important issue pursue theoretical interest practical consideration acknowledgements liu work supported university hong kong research committee research assistant professor scheme grant rgc hong kong grf project cao work supported part european research council netherlands organization scientific research persis work partially supported dutch organization scientific research nwo auspices project quantized information control formation keeping quick stw perspectief program robust design systems auspices project cooperative networked systems hendrickx work supported belgian network dysco dynamical systems control optimization funded interuniversity attraction poles program initiated belgian science policy office references arenas kurths moreno zhou synchronization complex networks physics reports cao morse switching sysstems control letters persis frasca robust coordination ternary controllers ieee trans automatic control agent systems ieee trans automatic control synchronization complex networks nonlinear dynamical systems world scientific singapore meng xie triggering approach leaderfollowing problems automatica xiao meng chen consensus switching networks integrators based edge events international journal control yang ren liu chen decentralized consensus linear systems general directed graphs automatica zhu jiang feng consensus systems general linear models automatica
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analyses aggregate fluctuations firm network based criticality hiroyasu graduate school simulation studies university hyogo abstract apr study examine difference size avalanches among industries triggered demand shocks rephrased control economy fiscal policy using model observed data obtain following results size avalanches follows power law mean sizes avalanches industries diverse standard deviations highly overlap compare simulation table actual policies compatible keywords aggregate fluctuation demand network firm production inventory control theory study conducted part project price network dynamics small medium enterprises undertaken research institute economy trade industry rieti authors thank institute various means support thank hiroshi yoshikawa hideaki aoyama hiroshi iyetomi yuichi ikeda yoshi fujiwara wataru soma yoshiyuki arata members attended internal seminar rieti helpful comments gratefully acknowledge financial support japan society promotion science graduate school simulation studies university hyogo kobe hyogo japan introduction giving stimulus firms prompting spillover effect way government affect economy includes purchasing goods services giving grants firms taxes governments consider fiscal policy important determinant growth currently analysis table considered strong tool predict spillover effect enables obtain single predicted value spillover effect caused stimulus however obvious obtain result prediction even use exactly volume used calculation nevertherless normally expect result substantially around prediction concern whether expectation correct main topic study size spillover effect around average prediction true propagation never amplified reduced firm networks however gabaix showed firm size distribution hypothesis breaks addition acemoglu pointed microeconomic shocks may lead aggregate fluctuations presence intersectoral linkages studies suggest stimulus spillover effect result proximity prediction words normal distribution usually assumed assumption doubtful study reveals demand shocks outside cause spillover effect use micro model invented bak employ observed data clear following points diversity spillover effect must depend industries shocks given extent getting involved spillover effect must also depend industries remainder paper organized follows section introduce dataset section describes methodologies utilize analyses section presents results finally section concludes data use datasets tsr company information database tsr company linkage database collected tokyo shoko research tsr one major corporate research companies japan datasets provided research institute economy trade industry rieti particular use dataset collected tsr data contain wide range firm tion necessary information study use identification capital industry type suppliers clients construct entire network firms based suppliers clients note suppliers clients firm data may considered constraint limits number links node however node suppliers nodes without limitation long clients designate node supplier vice versa therefore numbers suppliers clients limited number firms nodes number ties links network direction direction important study split firms based industries industries classified japan standard industrial classification mainly use division levels classifications however make alterations classification since classifications government except elsewhere classified industries unable classify less important study omit addition separate whole sale retail trade wholesale retail difference divisions negligible study shocks outside fiscal policies often occur retail therefore division level study alterations shows industries moreover use three industries group level compare effects japanese fiscal policies groups new motor vehicle stores electrical appliance stores except secondhand goods real estate agents difference division group levels clear later industries concern confusing two levels use table compare prediction size micro model table closest table time use updated table figure shows degree distribution observed network red plots distribution observed network important point distribution means distribution decay seems fit plots line cumulative probability degree positive constant degree distribution normal distribution plot shaped blue plots figure since normal distribution exponentially decays observe blue plots decrease plot create random network explained section reason compare observed network random network random network creates aggregate fluctuation decays exponentially show section words tells random network network real economy aggregate fluctuation however observed network random network see figure probability distribution cumulative probability distribution fitted line said distribution distribution network distribution often called network pointed nature networks determinant aggregate fluctuations since observed network network expect aggregate fluctuations network methodology use modified model based production model modified model enables conduct simulations investigate characteristics aggregate fluctuations model production inventory originally invented bak model assumes firms connect supply chains firm amount inventory firm receives orders clients supplies intermediate goods services clients firm enough inventory sends orders suppliers therefore cascades orders production sometimes occur size cascades defined total extent production due activated firms bak showed distribution cascade size follows power law result underlies recent studies related aggregate fluctuations cascade reaction understood aggregate fluctuations brevity call cascade reaction avalanche result obtained bak strongly depends regularity supply chain network node two suppliers two clients regular network except nodes top bottom layers already shown real network regular network network assumption strong apply model real network mitigate limitation regular network iino iyetomi generalized model node arbitrary numbers analyzed nature generalized production model employ generalized model minor modification describe model used analyses every time step every firm new amount inventory decided based following equation amount inventory firm time amount orders received firm time amount production conducted firm time equation renews inventory depicted figure assume firm equally sends orders suppliers firm produces one unit production one unit material obtains firm produces minimum goods necessary meet requests consumers assumptions simply result production feature multiple number suppliers firm addition assumption results based inventory renewal equation assumptions amount production given number orders firm places supplier calculated ceiling function since quantity received orders sum orders firm one clients firm firm client firm needs produce firm regarded belonging primary industry assumed able produce arbitrary amount production figure first orders placed outside depending analyses firm selected firms firms specific industry place order selection uniformly random two firms may mutually supply long step supply chains may form cycle possible firms loop indefinitely produce goods services although never occurs real economy iino iyetomi assume firms randomly assigned potential values analogy electrostatics firm potential another supply similar water flow although assumption helps avoid loop useful analyzing nature randomly created networks particularly clear assign value firm make simple assumption firm already supplied products firm already propagation process ignored supplier figure shows example loop three firms firm ignores firm needs production precisely supply link firm firm tentatively ignored since observed data include industries may considered unnatural contemplate inventory service industries understandable consider inventory intangible products insurance healthcare however service industries inventory service ready used considered inventory example vacant hotel room ready use incurs cost therefore discuss industries deal tangible intangible products network results section show nature avalanches diversity industries start results avalanches comparing random network observed network random network every pair nodes connected according constant probability expected number links random network number nodes random network created network number nodes number links observed network observed network nodes links therefore set approximately result obtain random network nodes links accordance necessary compare two networks experiments proceed follows networks time firm randomly chosen firms chosen firm sells unit product avalanche calculated repeat billion times proceeds billion avalanche size aggregated production obtain every time step figure shows avalanche sizes two networks red plots observed network blue plots random network random network obviously decays fast seems distribution fitted line hand observed network part fitted line result network avalanche size already shown analytically proved partly constraints figure results tell uniformly random stimuli cause avalanche real network average avalanche size representative value table analysis results single value prediction means representative value used predict spillover effect however distribution typical scale seems careful analyses required table viewpoint fiscal policy important know stimulus received different industries causes differences conduct experiments changes previous experiments firm randomly chosen industry industry fixed experiment every experiment billion instances demand given experiments conducted industry divisions figure shows distributions avalanche sizes expected distributions avalanche sizes would different shapes turns untrue observed apparent difference shapes figure figure shows mean sizes avalanches however mentioned last paragraph size distribution therefore average representative avalanche size error bars figure show standard deviations since already observed avalanche size power law distribution figures know standard deviations variances large conduct statistical test difference average billion samples cause small standard errors always show significance difference therefore test pointless instead important standard deviations overlapping seems strongly expect spillover effect started specific industry certainly superior inferior industries figure figure order industries aligned horizontal axis roughly shows advancement primary industries since advanced industry manufacturing services downstream supply chains long chains primary industries may considered distance primary industries positive correlation coefficient avalanche size industry however observe relationship attributed network structure network network firm short path hub since hub yields large magnitude avalanches seems dominated mechanism may argued production model study model doubtful much explain actual economy therefore simply compare size avalanches inverse matrix table pearson correlation coefficient consider production model include data trade volume said coefficient surprisingly large noted production model simulate variances obtained table conduct experiments start group level industries compare effect actual policies results figure based division level industries industries new motor vehicle stores carsale electrical appliance stores except secondhand goods electronicssale real estate agents brokers housesale group level correspond target industries past japanese fiscal policies tax breaks system housing system system home electronics setup experiments previous experiments figure since government publishes actual size budgets institutes publish estimated economic results validate predictability model government lost tax revenue corresponding billion japanese yen billion dollars assumed exchange rate japanese yen dollar tax breaks economic result estimated billion japanese yen billion dollars however estimation include indirect effects industries leverage return investment government spent billion japanese yen billion dollars system housing total economic result estimated billion japanese yen billion dollars leverage government spent billion japanese yen billion dollars system home electronics total economic result estimated trillion japanese yen billion dollars leverage although economic result leverage estimations abovementioned studies system housing apparently small find experiments three samples small find apparent contradiction results observe distributions avalanche sizes stated specific industries figure shapes tails similar interpreted certain supply chain always used large avalanches supply chain may lie particular industries examine hypothesis obtain different measure first experiments measure often firms industry become involved avalanches note examine avalanche sizes start specific industries thus far firm randomly selected firms surprisingly observed figure extent firm becomes involved avalanches sharply different sharpness totally different observed figure wholesale manufacturing distinctly large construction included largest group result means firms industries apparently always involved large avalanches start industry figure conclusion paper analyzed cause diversity spillover effect used observed data transactions japan data employed production model result confirmed size spillover effect triggered demand follows power law therefore normal distribution usually expected analyses tables reliable assumption although use volume trade results simulations show significant correlation coefficients moreover simulated avalanche sizes policies actually conducted correspond estimation given evaluations policies addition industries diverse potential become involved avalanches references easterly rebelo fiscal policy economic growth journal monetary economics romp haan public capital economic growth critical survey perspektiven der wirtschaftspolitik leontief quantitative input output relations economic systems united states review economics statistics gabaix granular origins aggregate fluctuations econometrica acemoglu carvalho ozdaglar alireza network origins aggregate fluctuations econometrica bak chen scheinkman woodford aggregate fluctuations independent sectoral shocks criticality model production inventory dynamics ricerche economiche ministry internal affairs communications japan standard industrial classification jsic summary development jsic eleventh revision trade ministry economy japan industry updated table iino iyetomi modeling relation transaction network production activity firms progress theoretical physics supplement zachariou expert takayasu christensen generalised sandpile dynamics artificial directed networks plos one watts small worlds dynamics networks order randomness princeton university press ministry internal affairs communications policy evaluation pertaining special taxation measures japanese shirai tax breaks cut total domestic emissions japanese japan center economic resarch monthly report new vehicle registration japanese http mizuho research institute evaluation system housing japanese board audit japan report system home electronics japanese figure comparison degree distributions random network observed network horizontal axis shows degree vertical axis shows cumulative probability blue plots indicate random network red ones indicate observed network figure generalized production inventory model scheme inventory renewal arrows show flows products therefore orders supplies opposite example loop avoidance assumption three firms firm asks firm supply products given firm inventory enough demand firm asks firm supply products however firm take supply firm link ignored figure comparison avalanche size distribution random network observed network horizontal axis shows sizes avalanches caused demand vertical axis shows cumulative probability blue plots indicate random network red ones indicate observed network although size zero avalanches ignore main aim figure show shapes tails zero included log plots figure comparison avalanche size distribution industries horizontal axis shows sizes avalanches caused demand repeated demand firm chosen randomly industry vertical axis shows cumulative probability although size zero avalanches omit aim figure show shapes tails zero included log plots figure comparison average avalanche size starting specific industry horizontal axis lists industries repeated demand firm chosen randomly industry vertical axis shows average avalanches error bars show standard deviations figure simulation industries compare fiscal policies horizontal axis lists industries correspond target industries past japanese fiscal policies tax breaks system housing system home electronics vertical axis shows average avalanches demand firm industry figure difference expectations involved avalanches horizontal axis lists industries vertical axis shows expectation involved avalanches per instance demand firm demand chosen randomly firms expectation industry averaged firms industry
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dynamic graph connectivity improved worst case update time sublinear space sep david gibb bruce kapron valerie king nolan thorn abstract paper considers fully dynamic graph algorithms faster worst case update time sublinear space fully dynamic graph connectivity problem following given graph fixed set nodes process online sequence edge insertions edge deletions queries form path nodes first data structure presented worst case time per operation polylogarithmic paper shave factor log time log per update sequences polynomial length algorithm answers queries log log log time correctly high probability using words size log matches amount space used graph connectivity streaming algorithm also show connectivity maintained using words amortized update time department computer science university victoria canada bmkapron val nolandthorn research supported nserc google faculty research grant introduction dynamic connectivity problem given undirected graph fixed set nodes online sequence updates queries following form delete delete edge insert insert edge query path nodes goal process graph updates queries efficiently possible data structure solves dynamic connectivity problem using worst case time per update operation log log log worst case time per query addition sequence updates polynomial length data structure needs sublinear amount storage namely words size log ensure high probability queries answered correctly way comparison dynamic graph stream problem process online sequence polynomial number edge insertions deletions stream ends spanning forest output goal minimize space algorithm first sublinear space dynamic graph algorithm matches space needed space efficient dynamic graph stream algorithm ahn guha mcgregor shaves factor log worst case time per update best previously known dynamic connectivity algorithm kapron king mountjoy application show first sublinear space dynamic connectivity algorithm uses amortized update time log log log query time words compares log word algorithm amortized update time log log log query time holm lichtenberg thorup sublinear space algorithm error constant keep list edges additional space error answer query yes answer always correct correct probability constant cases assume adversary knows edges graph constructs sequence updates queries knowledge random bits used algorithm whether queries answered correctly edges spanning forest used data structure revealed adversary several dynamic graph problems easily reduced problem maintaining connectivity see thus algorithm yields dynamic graph algorithm determining weight minimum spanning tree graph different edge weights cost updates increased factor also used maintain property bipartiteness constant factor increase cost see overview techniques nearly every dynamic connectivity data structure maintains spanning forest dealing edge insertions relatively easy challenge find replacement edge tree edge deleted splitting tree two subtrees replacement edge edge reconnecting two subtrees words cutset cut one subtrees simplify notation confusion allow denote set nodes tree otherwise use notation compare techniques paper dynamic graph connectivity dynamic graph stream paper three papers use random sampling identify name edge crosses cut dynamic graph paper introduced cutset data structure maintains dynamic forest disjoint trees subgraph graph tree leaving edge edge exactly one endpoint tree paper uses log words implement cutset data structure ensures edges leaving identified high probability edge inserted flips fresh coins determine samples appears records outcomes table graph stream paper implicitly uses restricted cutset data structure accommodates continuous stream nontree edge insertions deletions one final batch change nontree edges tree edges point leaving edge tree determined constant probability streaming paper uses log words implement restrictive cutset data structure words overall uses hash functions rather independent random bits determine whether edge sample eliminating need table remember sampling outcomes uses technique known sampling recovery sample find edge constant probability verify high probability edge cutset data structure also determines edge leaving constant probability using log words seen using sampling novel possibly practical method recovery three papers use log tiers cutset data structures build spanning forest similar parallel minimum spanning tree algorithm tier contains forest forest subforest tier cutset data structure tier leaving edges discovered become tree edges forest tier streaming paper needs form forests end stream worst case single insertion deletion cause superpolylogarithmic number changes edges leaving various tiers thus maintaining spanning forest update streaming algorithm would produced stream ended update costly terms time using high probability find edges leaving dynamic graph paper avoids problem finding edge leaving every tree tier extra cost log factor time space algorithm avoids cost using high probability necessarily keep tree edges streaming algorithm ensures number trees merged next tier induces subtle change invariants less restrictive related work related work dynamic connectivity see organization section define cutset data structure section show used solve dynamic connectivity section discuss implementation cutset data structures section give space efficient dynamic connectivity algorithm cutset data structure definition results consider forest disjoint trees subtrees graph dynamic undergoing updates form edge insertions deletions addition edge links two trees may updated become tree edge would cause two trees joined one similarly tree edge may changed edge problem design data structure goal maintaining tree edge cutset successful search returns edge cutset null otherwise assume graph updates oblivious randomized outcomes search may consider sequence updates fixed advance updates revealed algorithm one time operations insert edge adds edge make tree edge adds make nontree edge removes delete edge removes edge removes search returns null edges cutset returns edge cutset otherwise lemma let constant cutset data structure graph forest worst case update time space log words cutset constant probability search returns edge cutset probability less returns edge otherwise returns null always returns null auxilliary list words keep track current edge set graph search returns edge probability always returns edge returns edge implementation cutset data structure discussed section fully dynamic connectivity algorithm theorem constant sequence polynomially many updates dynamic connectivity algorithm answers queries correctly high probability uses time per update words error yes may output correct answer vice versa probability less list current edges graph maintained additional words sequence length updates may processed high probability single query correct case error answer query yes answer always correct may incorrect probability tiers invariants recall parallel algorithm building minimum spanning tree may view algorithm constructing sequence forests call tiers tier set nodes graph tier subforest minimum spanning forest formed tree picks minimum cost edge linking another tree algorithm similar structure tier top top log maintain cutset data structure forest refer collection cutset data structure generated using independent randomness tier top used form forest higher tier data structure consists forest ftop randomness array tier tier top search refers search operation cutset data structure similarly extend definitions operations cutset data structure tier since randomness array tier top delete edge top remove ftop appears nothing else graph updated cutset data structure tier used find edges leaving nonmaximal trees fragments tier given edge let minimum tier first appears refer tier edge maintain following invariants let current graph vertices every tier top let node tier top search successful properly contained node tier discussion note invariants necessarily preserve property every edge spanning tree ftop currently edge leaving fragment rather invariant ensures fragment tier search successful edge leaving also invariants preclude tree edge incident fragment search successful edges forests may thought historical search edges edges returned search gives flexibility perform updates quickly markov inequality implies constant probability constant fraction executions search fragments successful lemma let defined lemma cutset data structure fnum fragments probability least number fragments search succeeds least fnum proof fnum fragments cutset data structure expected number fragments search fails succeed greater since probability least finding tree edge markov inequality probability fnum fail find tree edge fnum edge leave two fragments easy see following observation let forest containing fragments set edges leaving least fragments spanning forest fragments lemma graph constant let top max invariants hold probability ftop spanning forest proof let constant lemma call tier fnum fragments successful number fragments search successful least fnum let tier successful otherwise lemma tier fnum fragments successful searches tier invariant fragments properly contained set tier edges includes edge leaving observation number fragments fnum since successful tiers suffice bring number fragments flip max coins probability heads expected number heads max probability number heads less less chernoff bound note results fixed graph extended hold poly arbitrary subgraphs including generated sequence polynomially many updates corollary let graph updates let constants assume top max cutset data structures spanning forest holds even probability every ftop dependence randomness pair proof consider collection graphs set lemma union bound fail spanning forest greater probability ftop performing updates say fragment tier isolated equal fragment tier efficiently maintain invariants edges updated check cutset induced tree invariants could become violated invariants maintained bottom changes tree may needed fragment becomes isolated tree edge deleted isolated fragment search may become successful fragment newly formed isolated let denote tree containing node forest edge inserted start lowest tier checking tier follows may affect search search isolated search successful edge returned added tree edge added must added higher tiers forms cycle highest tier edge cycle made edge may turn cause isolated fragment higher tier search must checked edge deleted affect results search fragment calls search potentially affected point proceed case insertions give algorithms handling updates queries following procedure restores invariants update involving edge described used insertions deletions initialize tier top initialize cutset data structure contains trees consisting single vertices insert top call insert edge insert call refresh delete top call delete edge delete call refresh query ftop return yes else return proof following given appendix lemma invariants maintained initialize insert delete algorithm refresh top isolated search returns edge path exists tier let lowest tier path let edge maximum tier path tier call insert edge call make tree edge top end end end end implementation details running time space analysis briefly review basic data structures used based observation tree linked another tree sequence given euler tour new tree obtained constant number links cuts sequences old trees sequence stored balanced ordered binary tree properties node graph value stored words length log following hold tree edge deleted inserted log time sum values stored nodes given tree returned time value node updated log time per word given node name tree containing returned log time name tree obtained quickly one keeps degree log solely purpose obtained log log log one willing spend time insertion deletion tree edge aka trees provide means finding maximum weight edge path two nodes dynamic forest log time per tree edge link cut find operation require space size vector value stored node edge implementation details ftop stored values associated node queries answered time log log log ftop also stored tier tree edges weight tree data structure allows find cycle edge log time worstpcase new tree edge found every tier twice changes filtered total top tree edge insertions deletions lemma let update time cutset data structure space running time space fully dynamic connectivity algorithm space needed top since top log lemma lemma imply theorem implementation cutset data structure illustrate main idea data structure first explain simplified deterministic version works exactly one edge cutset let current graph edge name bit vectors containing binary representations nodes respectively denotes bit vector followed say edge given node maintain vector equal bitwise xor sum mod vectors assigned incident edges use summation notation denote bitwise xor two vectors tree maintains bitwise xor vectors stored nodes use addition subtraction vectors refer bitwise xor vectors operations implemented follows insert edge add name vectors nodes make tree edge insert linking representing make nontree edge delete removing contains delete edge subtract vector vectors nodes edge remove calling make nontree edge search let sum vectors nodes return lemma tree exactly one edge cutset cut search always successful amount space used log bits proof every edge two endpoints tree added vectors two nodes contributes twice thus appears sum every edge exactly one endpoint contributes exactly cutset exactly one edge name edge extension cutsets one edge extend data structure handle case cutset one edge may sublinear space require assumption edge deleted unless currently graph number updates bounded constant alternatively keep list edges currently graph maintain node array indexed levels leveln let denote vector level node edge inserted leveln probability add idea size cutset constant probability exactly one edge cutset unique name edge added case name one edge search successful decide edges included sums various levels without recording coinflip use independent hash function mapping edges denotes edge sampled level leveln cutset data structure iff following lemma straightforward observed previously proof appendix lemma let cutset probability integer exactly one hashes value verify high probability actually name one edge cut rather sum several names make use odd hash function say random hash function given set odd number elements hash probability mod use construction create hash function let pick uniformly random odd multiplier threshold two components define follows mod otherwise idea verification follows use independent function randomly partition set two test part using using odd hash function determine parts least one element repeating log tests fails high probability one element set particular letting algorithm initialized choosing following functions independently random independent hash function independently random hash function used determine tag edge particular level node maintain auxilliary vector tag edge added bitwise may view verification problem instance recovery possible apply known techniques problem setting particular proposes approach uses polynomial evaluation modulo prime verify uniqueness forprecovered values set size success probability method maintains sums form weight item randomly chosen using approach purposes order search maintain correct values probability sequence edge updates must work adding edge would require adding mod term best knowledge requires performing roughly multiplications numbers reduction modulo alternatively independent parallel repetitions technique primes size bits contrast approach may done multiplications random numbers reduction modulo power vector edge name added level sums nodes tree auxillary vectors maintained well sums vectors let specifics described operations make tree edge make nontree edge implemented case cutsets size one remaining operations implemented follows insert edge leveln add sample add name value add tag add delete edge leveln name value subtract edge call make nontree edge remove containing search leveln edge cutset found test follows let edge given minimum pair edge cutset returned else null returned let sum level vectors tree minimum level constant probability name edge cut name edge high probability pair must least one edge cut since otherwise edges one edge cut let two edges let since independent probability therefore probability recall independently chosen random hash functions therefore given probability probability equality fails pairs less sufficiently large therefore search returns edge edge cut probability correctness stated lemma follows running time cutset data structure hash function requires constant time perform hash log hashes performed determine edge sampled level computing tag node tree forest cutset data structure vector log words since log levels level two log bit vectors maintaining sum vectors log vectors size log requires time similarly make nontree edge make tree edge requires time search involves finding maximum level checking tag determining current edge list determining name trees containing endpoints take log time hence conclude update time edge insertion deletion time search time log concludes proof lemma section treat dynamic connectivity algorithm two instances space efficient dynamic connectivity data structure presented previously black boxes denote respectively generated using independent randomness using black boxes present algorithm solve dynamic connectivity words query time update time within log factor amortized update time log log log query time idea maintain certificate connectivity using dynamic connectivity method shown previous sections use certificate input algorithm give definitions dynamic graph maximal forest maintained graph forest edges removed maximal forest maintained graph input connected forest maintained present algorithms handling updates queries insert insert process updates edge added delete edge removed insert process updates edge added insert edge removed remove delete edge insert except first step replaced delete query query see connected theorem fully dynamic connectivity maintained amortized time per update log log log per query using space words polynomial length sequence updates proof given appendix references kook jin ahn sudipto guha andrew mcgregor analyzing graph structure via linear measurements yuval rabani editor soda pages siam otakar jistem problemu minimalnim certain minimal problem czech german summary prace mor graham cormode donatella firmani unifying framework algorithms distributed parallel databases monika henzinger valerie king randomized fully dynamic graph algorithms polylogarithmic time per operation acm july jacob holm kristian lichtenberg mikkel thorup deterministic fullydynamic algorithms connectivity minimum spanning tree biconnectivity acm bruce kapron valerie king ben mountjoy dynamic graph connectivity polylogarithmic worst case time sanjeev khanna editor proceedings annual symposium discrete algorithms soda new orleans louisiana usa january pages siam mikkel thorup sample distinguisher probability corr appendix additional proofs lemma invariants maintained initialize insert delete proof clear invariants hold call initialize consider update operations maintenance invariant holds trivially refresh never makes edge tree edge tier invariant first note refresh checks cycle removes edge potential cycle inserting edge causes cycle tree forest inclusion property first note handling deletions edge deleted every tier violate invariant insertions deletions line refresh makes tree edge tiers inclusion immediately maintained cycles created suppose edge would form cycle inserted let edge maximal tier cycle first note due fact isolated minimality maximality tier must case edge forest tier therefore removed forests containing inclusion property invariant maintained invariant fails isolated fragment tier search successful call bad initially start empty graph bad assume first update bad created show contradiction bad created either newly formed update update search successful current graph two subcases isolated update isolated update search became successful update let updated edge first show claim call refresh bad fragment introduced tier form remains case execution refresh proof claim call refresh two ways update could create bad fragment first may occur insertions deletions happens fragment tier search return edge update case cut induced changed exactly one node second happen refresh deletions particular removed case two new fragments created cuts changed hence claim proved bad fragments created call refresh consider happens refresh execution refresh bad fragment may result made tree edge tier line causing two fragments joined one new fragment may bad since result search search new fragment contains invariant every tree higher tiers created inserting note replacing one edge another edge cycle affect cut replacement cause creation new bad fragment concludes proof claim given claim show calling refresh invariant holds proof induction value refresh assume top suppose iteration invariant holds tiers still case iteration since iteration makes changes tiers suppose bad means isolated search returns edge since inserted tier longer bad invariant holds tiers argument applies maintenance invariant holds induction top lemma follows immediately following lemma probability integer exactly one hashes value proof let prove statement lemma evaluate inside summation independence theorem fully dynamic connectivity maintained amortized time per update log log log per query using space words polynomial length sequence updates proof first prove correctness let spanning forest spanning forest certificate connectivity connected correctness depends whether correctly maintained spanning forests respectively proof theorem ftop maintained spanning forest since uses independent randomness updates therefore updates input graph independent randomness similarly ftop maintained spanning forest union bound forests maintained correctly note always correct observe proof theorem single update results log updates ftop turn result updates update amortized time hence total time per update worst case time plus worst case time plus amortized time total amortized update time space used log words space used words total space used words
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nov using noisy extractions discover causal knowledge dhanya sridhar university california santa cruz dsridhar jay pujara information sciences institute jay lise getoor university california santa cruz getoor introduction knowledge bases constructed information extraction text play important role query answering reasoning since automatic extraction methods yield results varying quality typically extractions retained ensures precision expense recall consequently noisy extractions prone propagating false negatives used reasoning however many problems empirical observations entities observational data readily available potentially recovering information fused noisy extractions reasoning tasks empirical observations extractions obtained open critical problem designing methods exploit modes identification work study particular reasoning task problem discovering causal relationships entities known causal discovery two contrasting types approaches discovering causal knowledge one approach attempts identify causal relationships text using automatic extraction techniques approach infers causation observational data example prior approaches mined causal links regulatory relationships among genes directly scientific text however extracted links often miss complex patterns require observational data hand given observations alone extensive work studied problem inferring network relationships among variables observational data gene expression measurements used infer causal relationships gene regulation prior approaches use constraints find valid causal orientations observational data although constraints offer attractive soundness guarantees need observed measurements variables remains costly prohibitive experimental data unpublished extractions interactions genes mined directly text provide coarse approximation unseen observational data combining extractions mined kbs observed measurements available causal discovery alleviate cost obtaining data propose approach fusing noisy extractions observational data discover causal knowledge introduce aus use probabilistic model causal relationships combines commonly used constraints observational data extractions obtained aus use uses probabilistic soft logic psl modeling framework express causal constraints natural logical syntax flexibly incorporates observational modes evidence main contributions introduce novel problem combining noisy extractions observational data conference neural information processing systems nips long beach usa propose principled approach uses causal discovery constraints recover patterns consistent predictions cheaply acquired extractions provide proxy unseen observations apply method gene regulatory networks show promise exploiting signals causal discovery suggesting critical new area research compare aus use conventional approach uses observational data perform causal discovery evaluate methods transcriptional regulatory networks yeast results validate two strengths approach aus use achieves comparable performance conventional method suggesting noisy extractions useful approximations unseen empirical evidence global logical constraints observational data enforce consistency across predictions bolster aus use perform par competing method results suggest promising new directions integrating knowledge bases causal reasoning potentially mitigating need expensive observational data background logical causal discovery inputs traditional causal discovery methods independent observations variables problem causal discovery infer directed acyclic graph dag edge eij corresponds direct cause changing value always changes value since graphical model encodes conditional independences among causal discovery algorithms exploit mapping observed independences data paths specify constraints output algorithm canonical method performs independence tests observations rule invalid causal edges constraints causal graph structure also encoded logic logical causal discovery system independence relations represented logical atoms logical atoms consist predicate symbol variable constant arguments take boolean continuous truth values avoid confusion logical variables remainder paper refer vertices inputs logical causal discovery require following predicates represent outcomes independence tests among ndep refers statistical dependence vertices measured independence test conditioning set empty set ond ond ndep corresponds statistical dependence vertices conditioned set independence test performed outputs logical causal discovery system represented following target predicates auses refers absence presence causal edge substituted pairs vertices finding truth value assignments atoms goal causal discovery ncestor corresponds absence presence ancestral edge vertices ancestor directed causal path may additionally infer truth values ancestral atoms jointly causal atoms given independence tests input goal logical causal discovery find consistent assignments causal ancestral output atoms using extractions causal discovery problem fusing noisy extractions causal discovery addition observations given set variables knn evidence knowledge base kij affinity score interaction based text extraction extending previous logical causal discovery methods additionally represent predicate set exta exta corresponds kab denotes absence presence undirected edge adjacency extracted text evidence adjacencies critical inference auses however adjacencies standard causal discovery inferred statistical tests alone approach replace statistical adjacencies exta goal fusing evidence logical causal discovery find maximally satisfying assignments unknown causal atoms based constraints independence signals section present probabilistic logic approach defining constraints using statistical evidence probabilistic approach inferring causal knowledge approach uses probabilistic soft logic psl encode constraints causal discovery key advantage psl exact efficient map inference finding probable assignments first review psl present novel encoding constraints combine statistical information probabilistic soft logic psl probabilistic programming framework random variables represented logical atoms dependencies encoded via rules logic logical atoms psl take continuous values logical satisfaction rule computed using lukasiewicz relaxation boolean logic relaxation continuous space allows map inference formulated convex optimization problem solved efficiently given continuous evidence variables unobserved variables psl defines following markov network called markov random field continuous assignments exp normalization constant max feature function scores configurations assignments linear function variable assignments defined psl model set weighted disjunctions rules weight rule rules consist logical atoms called ground rules constants appear atoms obtain first substitute logical variables appearing constants observations producing ground rules observe truth values subset ground atoms infer values remaining unobserved ground atoms ground rules corresponding weights map derive lukasiewicz relaxation applied ground rule derive hinge penalty function violating rule thus map inference minimizes weighted rule penalties find minimally violating joint assignment unobserved variables arg min max psl uses consensus based admm algorithm perform exact map inference aus use aus use extends constraints introduced algorithm whereas infers adjacencies conditional independence tests aus use uses adjacency evidence causal constraints adjacency evidence bridges contained kbs statistical tests propagate causal information figure shows rules used aus use first set rules follow directly three constraints introduced additionally introduce joint rules induce dependencies ancestral causal structures propagate consistent predictions describe aus use rules upgrade combine statistical signals causal discovery figure psl rules combining statistical tests evidence causal discovery rule type rules joint rules rules exta auses auses auses exta exta exta ond set auses exta exta exta ond set auses auses ond indep set exta auses auses auses exta auses auses auses exta auses exta exta ond indep set auses auses rules uses conditional dependence adjacency rule violating causal orientations however aus use adjacencies directly mined rule discourages causal edges vertices adjacent based evidence text rule penalizes simple cycles two vertices rules capture first rule orient chain based independence criteria rule orients path avoid orienting additional rule maps third rule orients avoid cycle applies rules iteratively fix edges whereas aus use rules induce dependencies causal edges encourage parsimonious joint inferences joint rules joint rules encourage consistency across ancestral causal predictions constraints transitivity follow basic definitions rule encodes causal edges also ancestral definition rule contrapositive penalizes causal edges nondescendants rule encodes transitivity ancestral edges encouraging consistency across predictions rule infers causal edges probable ancestral edges adjacent based textual evidence rule orients chain diverging path likely ancestor joint rules give preference predicted structures respect ancestral causal graphs evaluation investigate implications using noisy proxy adjacency benefits joint modeling experimental evaluation experiments investigate two main claims approach study whether noisy extractions suitable proxy latent adjacencies give similar performance conventional approach impute adjacency values using observations understand role joint ancestral causal rules observational data mitigating noise evidence evaluate aus use gene regulatory networks yeast compare psl model variant performs prototypical causal discovery using observational data aus replaces exta tandarda adjacencies computed conditional independence tests data dataset evaluation consists transcriptional regulatory network across genes yeast simulated gene expression challenge snowball sample smaller subnetworks sizes low jaccard overlap perform cross validation data contains gene expression measurements simulated differential equation models system perform independence tests measurements known contribute numerous spurious correlations addition gene expression data model domain knowledge based undirected interaction ppi edges extracted yeast genome database ocal ppi auses obtain affinity scores interaction pairs yeast genes string database string finds mentions gene protein names across millions scientific articles computes mentions genes additional step string extracts relations genes increases affinity score genes connected salient terms binds results model variant aus use aus aus uses aus table aus use achieves comparable performance aus suggesting noisy extractions approximate unseen adjacencies without joint rules aus uses shows worse performance pointing benefit sophisticated joint modeling mitigating noisy extractions evaluate aus use aus using cross validation networks uses rules approach computes tandarda ground atoms never appear groundings ond ndep based definition evaluate additional benefit joint rules compare aus use run causal orientation rules denoted aus uses aus respectively table shows average scores model variants regulatory network prediction task noisy extractions maintain performance first see comparable performance aus use aus answering first experimental question closely noisy extractions approximate adjacencies table statistically significant difference scores aus use aus comparable performance aus use suggests noisy extractions substitute observational data computations without significantly degrading performance joint rules overcome noise investigation model variants sheds light second experimental question around logical rules overcome noise extractions comparing variants method aus gains aus uses suggesting sophisticated joint rules needed mitigate noise extractions consistency across predictions encouraged joint rules bolsters adjacency signal extractions yield higher precision lower recall investigate extraction evidence mined string compare tandarda exta adjacencies obtain undirected regulatory links table shows average precision adjacency precision recall exta tandarda table adjacencies achieve lower recall substantiating need joint rules recovering missing causal orientations recall adjacency evidence type across subnetworks interestingly exta achieves higher precision statistical counterpart however tandarda gains recall result substantiates benefit joint modeling recovering additional orientations inputs nonetheless comparison points need deeper understanding role kbs play causal reasoning experiment details obtain marginal conditional dependence tests use linear partial correlations fisher transformation condition sets size two set rule weights psl models except rule set since encodes strong acyclicity constraint models use threshold categorize independence tests ond ond ndep ndep select cross validation hold subnetwork turn use best average score across subnetworks pick raised powers aus selects two different values binning independence tests computing adjacencies aus use requires single tests also select rounding thresholds psl models within framework since typically small rescale truth values ond ndep ndep reduce rightskewness values rescale string affinity scores related work work extends methods causal discovery notably algorithm first infers adjacencies maximally orients using deterministic rules based conditional independence supports external evidence form fixed edges nonedges work motivated recent approaches cast causal discovery sat instance conditional independence statements approaches based logical representations readily admit additional constraints relations domain knowledge however far logical causal discovery methods use external evidence identify probable edges separate vein prior work extended identify regulatory networks genetic interactions scientific literature contrast goal propose techniques leverage statistical test signals text evidence work similar combines gene expression data evidence mined knowledge bases infer gene regulatory networks however regulatory network inference orients edges using knowledge transcription factors instead reasoning causality approach propose principled causal discovery formulation basis incorporating evidence discussion future work work present initial approach reasoning noisy evidence directly logical causal discovery system benefit flexible logical formulation supports replacing conventional adjacencies computed observational data cheaply obtained extractions evaluation suggests noisy proxy signal achieves comparable performance conventional methods promising result points future research exploiting kbs causal reasoning greatly mitigating need costly observational data see many directions future work including better extraction strategies mining scientific literature finding proxies additional statistical test signals kbs could provide ontological constraints semantic information useful causal reasoning additionally plan study constraints causal discovery references stephen bach matthias broecheler bert huang lise getoor markov random fields probabilistic soft logic journal machine learning research jmlr appear david maxwell chickering learning bayesian networks learning data pages springer panagiotis chouvardas george kollias christoforos nikolaou inferring active regulatory networks gene expression data using combination prior knowledge enrichment analysis bmc bioinformatics tom claassen tom heskes logical characterization causal discovery uai antti hyttinen patrik hoyer frederick eberhardt matti jarvisalo discovering cyclic causal models latent variables general procedure uai antti hyttinen frederick eberhardt matti causal discovery conflict resolution answer set programming uai sara magliacane tom claassen joris mooij ancestral causal inference nips daniel marbach robert prill thomas schaffter claudio mattiussi dario floreano gustavo stolovitzky revealing strengths weaknesses methods gene network inference proceedings national academy sciences hoifung poon chris quirk charlie deziel david heckerman literome genomic knowledge base cloud bioinformatics robert prill daniel marbach julio peter sorger leonidas alexopoulos xiaowei xue neil clarke gregoire gustavo stolovitzky towards rigorous assessment systems biology models challenges plos one carlos heladia salgado irma julio automatic reconstruction bacterial regulatory network using natural language processing bmc bioinformatics song chen text mining biomedical literature constructing gene regulatory networks interdisciplinary sciences computational life sciences peter spirtes clark glymour algorithm fast recovery sparse causal graphs social science computer review damian szklarczyk john morris helen cook michael kuhn stefan wyder milan simonovic alberto santos nadezhda doncheva alexander roth peer bork string database association networks made broadly accessible nucleic acids research
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beyond sift using binary features loop closure detection sep lei guyue lan paper binary feature based loop closure detection lcd method proposed first time achieves higher performance compared sift feature based approaches proposed system originates previous work hashing loop closure detection mild employs hashing mih approximate nearest neighbor ann search binary features accuracy mild limited repeating textures inaccurate image similarity measurement burstiness handling introduced solve problem achieves considerable accuracy improvement additionally comprehensive theoretical analysis mih used mild conducted explore potentials hashing methods ann search binary features probabilistic perspective analysis provides freedom best parameter choosing mih different application scenarios experiments popular public datasets show proposed approach achieved highest accuracy compared running databases containing thousands images introduction loop closure detection serves key component globally consistent visual slam systems various approaches proposed address problem either suffer low accuracy perform unstably different scenarios low efficiency take many computational resources find loop closure candidate binary feature orb brisk based methods benefit low computational complexity efficient memory storage requirements also suffer low precision instability showing wide gap accuracy compared feature sift surf based approaches previously work mild first binary feature based method achieves comparable accuracy performance employing mih ann search binary features instead conventional methods rely bow scheme however accuracy mild still inferior realvalued feature based approaches worthwhile doubt whether ceiling binary feature based lcd approaches reached whether algorithms perform better binary work supported part natural science foundation china nsfc contract lei han lan robotics institute hong kong university science technology clear water bay hong kong lhanaf lxuan guyue zhou shenzhen china fang dept automation tsinghua university beijing china methods terms accuracy studied problem context reconstruction argue popular binary features orb brisk inferior sift based method terms reconstruction accuracy completeness without significant better computational performance would argue binary features different characteristics treated differently exploit potential binary features lcd approaches procedure mild binary feature orb based lcd approach achieves better accuracy sift based methods presented support idea one big challenge mild achieving high accuracy inaccurate image similarity measurement two images shown fig taken different places sharing high similarity score elements windows wall tiles existing different places tend cause confusion decrease accuracy tracing source find binary features less discriminative compared features since binary features carry much less information burstiness phenomenon severe binary features leads high similarity score images taken different places inspired propose burstiness handling procedure overcome misleading cases extra computational burden introduced procedure employing specific features mih mih important tool ann search binary features mild analysis mih incomplete search radius substring hashing procedure limited lack immediate probabilistic conclusions shown techniques may lead better performance considering accuracy efficiency locality sensitive hashing lsh sift features however theoretical analysis performance gain provided case binary features propose comprehensive analysis mih allowing arbitrary positive integers using based extension effect lsh binary features presented clearly shown fig even though mih efficient filtering dissimilar features considerable part candidates selected using mih would still outliers large datasets robust early termination strategy would reduce frequency memory access well hamming distance calculation without loss accuracy summary contributions paper include comprehensive theoretical analysis mih ann search binary features presented enlarging image taken left image taken right camera camera image taken left image taken right camera camera fig image captured different places shared high similarity score search radius substring hashing procedure limited zero original mild improved accuracy efficiency higher accuracy achieved due dealing burstiness similar features higher efficiency due use early termination omit features selectively based probabilistic analysis remarkable designed burstiness handling technique coped mih procedure extra computational burden introduced procedure sparse match proposed frame matching using binary features times faster traverse search average precision shown sec related works various approaches proposed loop closure detection either using global image signatures local image features global signatures relatively compact measure similarity two images efficiently computationally expensive extract global signatures thus unpractical visual slam applications recently features also shown potentials lcd ability extracting semantic information images paper focus local feature based approaches aiming provide efficient robust loop closure detector visual slam applications local features orb sift widely used visual slam systems frame registration proven accurate robust various scenarios local feature based lcd approaches mainly explore two feature classification image similarity measurement feature classification tries cluster different features indicating place image similarity measurement reveals possibility two frames indicating place based previous observations previous methods rely bow scheme efficiency considerations bow features clustered different centroids visual words using trained dictionaries frames represented histogram visual words loop closure likelihood calculated based difference visual word histograms however least two problems exist methods perceptual aliasing features clustered visual word may indicate different locations high complexity features low accuracy binary features among methods achieves highest accuracy performance terms recall precision takes per detection first tries binary features bow scheme called bag binary words able handle images much lower accuracy mild argue inefficiency bow binary features unlike mild using mih replace bow simply project binary features space feasible bow scheme however transformation hamming distance euclidean distance highly nonlinear training procedure required choosing projection functions limits range applications proposed system various methods proposed improve image similarity measurement including wgc burstiness improves image retrieval quality employing rotation scale information local features handles burstiness phenomenon visual elements assigning weights features based similarity others however procedure requires complex preprocessing remove repeating features image paper show combining characteristics mih burstiness handled barely computational burden iii loop closure detection noted previous work mild proposed lcd employing mih ann search binary features achieves significantly better performance compared binary feature based lcd approaches however accuracy mild still limited repeating textures common phenomenon artificial natural sceneries exploration mih ann search binary features inadequate search radius substring hashing procedure limited completeness provide brief review mild firstly based framework mild theoretical analysis mih ann search binary features explored extended sec long binary feature descriptor fin divided disjoint substrings shown fig features fall hash entry least hash table considered nearest neighbor candidates fin let denote collection nearest neighbor candidates fin thus image similarity measurement approximated fin fjk fjk fig framework mih binary feature fin divided disjoint substrings substring hash index kth hash table image index feature index stored corresponding entries reference feature fin considering mih next precise model image similarity measurement provided sec iiic improve accuracy lcd finally techniques early termination discussed sec improve efficiency mild assumption throughout paper hamming errors two feature descriptors evenly distributed used analysis mih review mild mild lcd divided two stages image similarity measurement bayesian inference image similarity used compute likelihood two frames loop closure bayesian inference employs temporal coherency get final probability loop closure based image similarity paper focus improving image similarity measurement calculated binary features directly hence accurate conventional methods using bow representation images query image firstly binary features extracted image similarity measurement denoted fip fjq refers binary feature similarity exp denotes hamming distance binary features fip fjq weighting parameter hamming distance threshold evaluates similarity two images using voting approach intuition behind eqn two images registered using binary features visual slam tend feature pairs small hamming distance results high similarity score image similarities candidate images stored database computed using mih evaluates performance mih two aspects complexity accuracy complexity indicates ratio features counted candidate set total dataset accuracy measured probability two features representing location encountered mih lower leads efficient image similarity computation higher indicates approximation accurate probability feature pair hamming distance encountered mih denoted precall function number hash tables search radius substring hashing procedure situations analyzed prior statistics hamming distance distribution orb features employed computing orb descriptor hamming distance distribution features location inliers different locations outliers approximated normal distribution respectively thus complexity accuracy mih computed precall precall precall best parameter mih chosen considering features substring fin selected performance mih features similar substrings fall candidate set discussed sec mih far approaches addressed problem fast search binary features using hashing technique including mih exact nearest neighbor search lsh ann search paper statistical information mih employed approximate nearest neighbor search lsh multiple substrings extracted original binary descriptor independently binary elements randomly selected original feature descriptor improve search precision maintain low complexity multiple hash table hashing strategies employed although extensive experiments conducted show hierarchical clustering tree hci performs slightly better lsh hci considered work randomized algorithm performance hci may unstable even dataset beside problems lcd random selection cluster centers possible features streamed database would impractical reorganize tree structure every time new feature enters mih resembles lsh methods divides long binary codes short substrings unlike lsh tries make substring independent mih takes inherent correlations substrings consideration mih feature pairs hamming distance less discovered sure hashing shown ability improve search performance lsh features adopted fast match binary features enlarging substring search radius however situations analyzed mathematical tools found paper complete analysis mih ann search binary features computing precall arbitrary nonnegative integer derivation implemented iterative way probability precall computed precall following precall equal probability independent balls thrown bins randomly least one bin balls situations analyzed without loss generality equals union two independent events least one bin ball bins one ball bin rest bins bin least one ball probability event equals precall computed event union indicates kth bin one ball rest bins least one ball based combinatorial analysis precall following probability event computed precall given similarly precall computed based precall larger given precall calculate complexity accuracy different parameter configurations shown fig concluded larger lead higher accuracy well higher complexity fixed achieve accuracy would efficient use smaller larger fig complexity accuracy different mih parameters including search radius hashing table number complexity accuracy grows monotonously fixed starts different shown figure additionally fixed overhead mih construction hash tables denoted ixed suppose hash tables mih hash entries indicates feature descriptor length orb features ixed analysis given required search precision find best parameter configurations mih minimize computational cost including complexity overhead ixed lcd problems minimum accuracy required expected larger three parameter candidates include loop closure detection problems attractive hash entries stored memory applications hash tables built would better choice complexity mih minimized need consider ixed implementations lcd chosen substrings represented short data structure efficient cpu operations burstiness handling discussed visual elements may appear times image statistically independent model would predict described burstiness phenomenon burstiness corrupts visual similarity measure context image search two types burstiness exist repeating texture exists scenery similar objects exist many places various weighting strategies proposed handle burstiness phenomenon paper combining framework mih handling burstiness without increase computational burden burstiness typically may take timeconsuming preprocessing procedure detect burstiness traversal feature extracted image following formulations sec probability independent features falling hash entry complexity approximated repeating features probability accuracy close configurations limiting number features falling hash entry image accomplish detection repeating features without extra preprocessing procedure burstiness weight feature similarity based total similarity score feature candidates provided mih given query feature fin feature similarity measurement modified fin log inverse document frequency term log represents total number candidate frames number frames similar feature fin repeating features may cause inefficiency features may little contribution image similarity measurement may exist candidate set frequently inefficiency solved limiting maximum number buckets nbuckets hash entry average number features falling hash entry would eatures maximum eatures eatures number features stored dataset substring length nbuckets set eatures experiment entries buckets nbuckets discarded early termination computational complexity mild grows linearly number images stored database large datasets part mild memory access feature descriptors hamming distance calculation given sparsity repeating locations found majority features candidate set outliers large hamming distances based observation adopt early termination avoid unnecessary memory access computational cost outliers exploiting partial information feature descriptor feature nearest neighbor candidate load first bits calculate partial hamming distance larger threshold regarded outliers directly instead loading rest bits experiments show early termination succeeds reject around outliers hamming distance larger experiments evaluate performance proposed approach conduct extensive different two individual experiments implemented lcd frame match experiments lcd reveal superior performance method loop closure detection experiments frame match verify precision proposed fast match algorithm sparse match experiments implemented ghz processor ram one core used compare computational efficiency proposed approach algorithms loop closure detection orb features extracted image using opencv feature descriptor length set newcollege contains images size citycentre contains images size images size images size bovisaoutdoor contains images size fig computational complexity proposed method terms running time frames newcollege dataset fig curve dataset loop closure detection accuracy evaluation curves datasets provided fig proposed approach achieves high accuracy datasets provided including indoor outdoor natural artificial sceneries particular present results newcollege dataset demonstrate reliability proposed approach shown fig nearly ground truth closures detected proposed method efficiency evaluation runtime lcd composed two main parts feature detection extraction image similarity measurement parts bayesian inference completed efficiently within thus omitted fixed given image resolution number features selected image grows linearly number frames stored database newcollege dataset frames average runtime procedure original version mild based proposed early termination technique reduced without influencing accuracy speed factor would even greater memory inefficient systems embedded chips fpga implementations time cost lcd frame also presented fig although running time grows linearly number candidate images efficiently handle loop closures datasets containing thousands key frames shown experiments enough visual slam designed applications indoor navigation systems fig accuracy distribution approximate nearest neighbour search using lsh sparsematch three nearest neighbour search methods used image matching sparsematch lsh method accuracy evaluated percentage correctly matched feature pairs valid feature pairs hamming distance less fig detected closure ground truth closure coordinate pixel represents index candidate image query image respectively red indicates detected closure approach yellow indicates ground truth closure brown indicates intersection detected closure ground truth closure red points neighbors brown points counted false positives rtabmap bowp dbow cnn feature ibuild verify performance proposed scheme compare work state art approaches rtabmap bowp based features well dbow ibuild mild use binary features used references comparison proposed system quantitative comparisons regarding accuracy recall rate precision runtime whole system including feature extraction loop closure detection shown table iva examining results presented proposed approach achieves highest recall rate nearly datasets sift surf feature based approach rtabmap ranks second place times slower proposed approach note bovisaoutdoor dataset proposed approach achieves much higher accuracy compared approaches similar features overexposure sun glare exist almost every frame causing confusion conventional lcd methods benefitting handling burstiness repeating texture negligible influence image similarity measurement frame matching benefit analysis performance mih ann search binary features sparse match proposed find corresponding features images frame matching applications overhead database construction must small high precision matching results required conventional methods based suitable applications require overhead initialization work repeating usage situations sparse match treated light version lcd mild proposed city centre new college indoor outdoor bovisa outdoor table comparisons algorithms terms accuracy recall rate precision efficiency average running time perframe stored database query image minimize overhead ixed maintain high precision parameter mih used sparse match chosen experiments image pairs consecutive images newcollege dataset orb features extracted image implemented verify performance mild terms accuracy efficiency experiments three image match methods implemented sparse match lsh implemented flann bruteforce match lsh choose parameters sparsematch hash tables bits key multiprobe level equal shown fig sparsematch achieved higher accuracy lsh average processing time image pair sparse match including overheads required search lsh experiments implemented computer number available cpu cores limited compare efficiency different algorithms conclusions binary feature based lcd approach presented paper achieves highest accuracy compared shown experiments running laptop higher accuracy achieved based handling burstiness reduces confusion caused repeating features complexity also reduced filtering outliers based partial hamming distance main bottleneck efficiency proposed system lies extraction binary features takes processing time fortunately computation shared binary feature based slam systems keep improving performance proposed lcd system maintain implementation currently proposed approach suitable datasets containing thousands keyframes larger datasets superiority effiency may decrease complexity mild increases linearly number candidates dataset memory management schemes combined enable proposed algorithm running constant time large scale problems better data structure suitable large hash tables may also adopted improve efficiency proposed lcd system eferences dan greene michal parnas frances yao hashing information retrieval foundations computer science annual symposium ieee raul jose maria martinez montiel juan tardos versatile accurate monocular slam system ieee transactions robotics vol dorian juan tardos bags binary words fast place recognition image sequences ieee transactions robotics vol mathieu labbe francois michaud loop closure detection online operation ieee transactions robotics vol ethan rublee vincent rabaud kurt konolige gary bradski orb efficient alternative sift surf international conference computer vision ieee stefan leutenegger margarita chli roland siegwart brisk binary robust invariant scalable keypoints international conference computer vision ieee sheraz khan dirk wollherr ibuild incremental bag binary words appearance based loop closure detection ieee international conference robotics automation icra ieee david lowe distinctive image features keypoints international journal computer vision vol herbert bay tinne tuytelaars luc van gool surf speeded robust features european conference computer vision springer nishant kejriwal swagat kumar tomohiro shibata high performance loop closure detection using bag word pairs robotics autonomous systems vol lei han fang mild hashing loop closure detection arxiv preprint accepted multimedia expo icme ieee international conference bin fan qingqun kong wei sui zhiheng wang xinchao wang shiming xiang chunhong pan pascal fua need binary features reconstruction proceedings ieee conference computer vision pattern recognition workshops matthijs douze cordelia schmid burstiness visual elements computer vision pattern recognition cvpr ieee conference ieee mark cummins paul newman probabilistic localization mapping space appearance international journal robotics research vol code avaliable https qin william josephson zhe wang moses charikar kai lsh efficient indexing similarity search proceedings international conference large data bases vldb endowment kiyosi introduction probability theory cambridge university press relja arandjelovic andrew zisserman vlad proceedings ieee conference computer vision 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parameterized approximation complexity detection pair problem graphs florent ralf jan january abstract study complexity problem detection pair detection pair graph pair sets detectors watchers listeners every pair vertices dominated watcher listener whose distances different goal minimize problem generalizes two classic problems dominating set metric dimension correspond restrictions respectively detection pair recently introduced finbow hartnell young finbow hartnell young complexity monitoring network watchers listeners networks accepted proved trees surprising result given dominating set metric dimension known solvable trees follows existing reduction hartung nichterlein metric dimension even bipartite subcubic graphs arbitrarily large girth detection pair approximate within factor parameterized solution size show using reduction set cover detection pair approximable within factor logarithmic number vertices input graph two main results algorithm fpt algorithm detection pair trees keywords graph theory detection pair metric dimension dominating set approximation algorithm parameterized complexity introduction order monitor faults network one place detectors nodes one possibility use local devices able detect location fault within distance one call watchers another kind detectors able determine exact distance fault device precise location call detectors listeners wish monitor network using watchers classic problem dominating set see books survey results problem hand want use listeners problem metric dimension see example papers references therein however useful use kinds detectors example one watcher enough monitor complete graph order would need listeners hand one listener would suffice monitor path order task would need watchers therefore finbow hartnell young recently proposed concept detection pair graph pair set watchers set listeners together monitor graph formally say vertex dominates vertices closed neighbourhood set neighbours together moreover vertex limos cnrs umr blaise pascal france labri cnrs umr bordeaux talence france separates two vertices distance different distance given pair sets watchers listeners say vertex distinguished either dominated watcher every vertex either dominated watcher separated listener detection pair every vertex distinguished size sum denoted note may choose place listener watcher position denote minimum size detection pair vertex called detector detection pair graph called resolving set smallest size resolving set called metric dimension denoted hand detection pair dominating set smallest size dominating set domination number denoted clearly min inequality strict follows graph without isolated vertices goal paper study decision optimization problems naturally associated notion detection pair computational problems mentioned paper defined formally section detection pair instance graph positive integer question detection pair pair instance graph task compute optimal detection pair algorithm given optimization problem algorithm returns solution whose size always times optimum refer book details decision problem parameter instance algorithm said fixed parameter tractable fpt short runs time computable function input size constant paper always consider solution size parameter refer books details finbow hartnell young proved detection pair trees pair solvable trees containing pair leaves common neighbour hardness result quite surprising since related problems set dimension general solved linear time trees see respectively note dominating set among classic graph problems see books metric dimension enjoyed lot interest recent years see papers paper continue study complexity detection pair pair initiated section describe reduction cover shows pair solved within factor logarithmic size input graph hand observe reduction hartung nichterlein dimension also applied pair implies pair approximate within factor sublogarithmic input graph order detection pair parameterized solution size hardness results hold even graphs bipartite subcubic arbitrarily large girth section prove pair linear time trees section show algorithm detection pair running fpt time log start paper preliminary considerations section conclude section paper log denotes natural logarithm function preliminaries start preliminary considerations definitions used computational problems formally define auxiliary computational problems used mentioned paper dominating set instance graph positive integer question dominating set set instance graph task compute optimal dominating set metric dimension instance graph positive integer question resolving set dimension instance graph task compute optimal resolving set set cover instance hypergraph positive integer question set cover vertex belongs set cover instance hypergraph task compute optimal set cover specific terminology graph vertex degree called leaf vertex called branching point degree least branching point special path starting ending vertex degree whose degree path called leg say attached note vertex belong leg special branching point leaves neighbours called given special branching point tree define subtree tree containing legs attached classic algorithm dimension trees use following results slater dimension trees classic literature metric dimension see also similar considerations proposition slater let tree set vertices containing special branching point legs attached leaves longest legs optimal resolving set following consequence proposition theorem slater dimension solved optimally linear time trees previous terminology branching points stems legs observations lemmas detection pairs following easy observations lemmas useful observation let graph set special branching points least two legs attached proof let detection pair show special branching point least one detector among vertices belonging leg attached indeed two neighbours belonging set would neither dominated watcher separated listener contradiction lemma let special branching point graph set leaf neighbours set legs length least attached let detection pair watcher least legs contain detector otherwise least legs attached contain detector proof watcher assume contradiction two legs contain detector neither listeners watcher distinguish two vertices distance contradiction similar argument holds vertices distance watcher lemma let graph special branching point leg attached whose leaf detection pair also detection pair proof let set vertices containing vertices leg attached whose leaf clearly pair vertices separated separated moreover vertices leg containing clearly distinguished vertex unique one distance completes proof general approximability section discuss general approximation complexity pair theorem pair approximated within factor log graphs order proof given graph build instance cover follows let set vertex pairs vertex set contains element dominates least one also set contains element therefore claim correspondance detection pairs set covers indeed every vertex set corresponds watcher placed vertex set corresponds listener placed vertex moreover set covers exactly elements correspond pair separated according detector placed since cover approximable within factor log polynomial time result follows hartung nichterlein provided reduction dominating set metric dimension reduction improved hartung phd thesis get reduction mapping instance instance bipartite graph maximum degree girth fact difficult see detection pair size detection pair size containing watchers words therefore aforementioned reduction also reduction detection pair hence hartung nichterlein state hardness approxmation factors log recent result inapproximability cover transfers set via standard reductions implies stronger statement theorem hartung nichterlein pair approximate within factor log instances order detection pair parameterized solution size let arbitrary integer results hold even instances bipartite maximum degree girth least algorithm trees section prove following approximability result theorem algorithm pair trees proof given input tree denote set vertices moreover define tree obtained follows vertex assuming legs length least attached remove leaf neighbours leaf neighbours iii leaf neighbours therefore least two legs attached vertex let describe algorithm compute set build compute optimal resolving set output first prove detection pair notice distances listeners vertices therefore pairs vertices distinguished listeners hand vertices dominated watchers therefore detection pair moreover clear computed linear time computed linear time using slater algorithm theorem therefore algorithm remains prove tree notice contains vertices part subtree special branching point indeed vertex special branching point special branching point also special branching point proposition optimal resolving set contains vertices belong subtree therefore suffices prove detection pair optimal resolving set every special branching point end let assume legs length least attached distinguish following cases case contains two detectors indeed contains one two listeners contains one watcher one listener lemma least one detector completes proof case case lemma detection pair detector least legs length least exactly listeners one watcher therefore detection pair done hence assume contains listeners precisely leaves belong detection pair done however lemma least detectors hence completes proof two infinite families trees show algorithm better approximation ratio first family consists trees built disjoint stars least three leaves whose centers adjacent additional single vertex solution set contains one watcher one listener every star simply let every center star watcher select listener hence approximation ratio second family consists trees built path vertices two add two leaves adjacent degree build star three leaves subdivide one edges moreover center made adjacent star contains two listeners two additional listeners selected among four degree adjacent put watcher center star add one neighbour endpoint listener approximation ratio fixed parameter tractable algorithm trees section provide exact algorithm detection pair fpt natural parameter solution size idea algorithm follows first search solution step may assume technical condition required subsequent steps algorithm proceed three phases first phase handle solution around special branching points tree see fixed set possibilities try branching point second phase determine set remaining listeners able compute set whose size bounded terms possible vertices may contain listener optimal solution finally third phase determine set remaining watchers able compute set vertices whose size bounded terms may used watcher algorithm checks validity possible choice placements theorem detection pair solved time log trees order preliminary lemmas describing algorithm prove series lemmas essential first lemma useful handle special branching points tree reducing problem around vertices fixed number cases lemma let graph special branching point least two legs optimal detection pair let set leaves adjacent let set legs length least attached denote vertices belonging leg let obtain another optimal detection pair replacing detectors one following sets detectors may use single watcher watcher listener leaf longest leg attached single listener leaf longest leg attached may use watcher listeners leaves legs watcher listeners leaves legs shortest one listeners leaves legs attached listeners leaves legs attached except shortest leg proof clearly watchers among may replace single watcher obtain valid detection pair distinguish two cases case contained unique detector watcher must dominated leaf neighbour since neighbour exists observed beginning proof may replace detector watcher case unique detector listener lemma must note listener seperates set pairs vertices hence may replace existing listener listener leaf longest leg attached separates least many vertices listener case done hence assume contained least two detectors using similar arguments one check solution case yields valid detection pair larger case observed replace watchers single watcher watcher let otherwise let set legs attached let lemma least legs contain detector assume first contained least detectors legs replace detectors listener leaf leg vertex leg distinguished distance listener placed leaf since listener separates set pairs obtained valid detection pair larger case suppose exactly detectors legs replace detectors listener leaf longest legs similar arguments vertices belong unique leg without detector distinguished listener separates set pairs hence vertex distinguished new detection pair must vertex separated vertex implies otherwise would separated listeners placed know separated detector must detector leg detector another leg could possibly separate since know one leg contained detector since shortest leg vertex distinguished contradiction hence valid detection pair whose size greater case done next two lemmas used algorithm choose placement listeners lemma let graph two vertices connected unique path let set vertices together vertices leg attached degree placing listeners vertices distinguished proof every vertex let note two vertices separated clearly distinguished let vertex assume contradiction vertex particular note every two vertices therefore one vertices belong assume belong shortest paths must say implies shortest paths also contradiction structure implies lemma let graph two vertices connected unique path either degree single leg attached moreover let set together vertices leg attached degree every detection pair one obtain valid detection pair replacing detectors two listeners proof lemma vertices distinguished note watcher could possibly distinguish vertex outside clearly distinguished furthermore since unique path connecting listener set pairs separated set pairs separated completes proof next lemma used algorithm determine placement watchers parts tree listener lemma let tree vertex optimal detection pair let subtree containing assume contains listener vertex shortest path goes particular contains listener iii maximal respect set size moreover determined time linear proof may view rooted tree whose root consider partition parts contains vertices distance since shortest paths listener vertex set listeners separates pair belong different parts therefore since detection pair two distinct vertices part least one dominated watcher let set parts size least parts least one vertex forest induced set vertices must dominated watcher let shall belong part together neighbours let prove satisfies claim first show assume contradiction case let watcher belongs optimality detection pair therefore must vertex dominated separated vertex dominated watcher words every listener thus since tree shortest paths common part say vertex disjoint iii maximal respect implies belong however observe belongs part otherwise would belong therefore uniquely determined within distance since implies fact distinguished contradiction thus shown moreover clear computed time linear using distances vertices therefore remains bound size contains vertices distinct parts part first show may associate two parts may vertices hence total number parts vertex since watcher may dominate vertices three parts part least contains vertices one vertex dominated watcher hence distinct parts leaves note leaf second prove either actual leaf vertex part claim contain leaves belonging part indeed leaf must dominated watcher dominated parent since parent dominate one leaf contains vertices parts size proves bound furthermore part since hence number leaves hence total leaves rooted forest levels leaves may size completes proof vertices hence algorithm proof theorem describe claimed fpt algorithm detection pair trees prove correctness running time proof theorem let describe algorithm seeks build detection pair detectors input tree order preliminary phase searching solution first step algorithm check whether solution set problem equivalent finding dominating set size done linear time set one may try possibilities placement unique listener possibility say placed listener vertex create subtree rooted follows first let except leaf attached may assume belongs similar argument proof lemma cases thus add remove leaves one attached one leaf keep apply lemma thus set vertices may contain watcher computable linear time suffices try possibilities selecting additional watchers set log possibilities check linear time whether valid detection pair total phase takes log time find valid solution phase return yes next assume proceed three phases phase handling special branching points preprocessing step using lemma first compute linear time set special branching points let set special branching points least two legs attached note observation return hence assume special branching point lemma assume four different choices set detectors vertices belonging leg attached therefore may possible combinations choices course discard choices detectors combination let partial detection pair corresponding remainder show decide whether detection pair phase determining set listeners let set vertices containing pair listeners vertices path puv connecting well vertices leg attached degree puv lemma vertices distinguished clearly subgraph connected subtree let set vertices neighbour vertex let txj subtree formed together trees containing neighbour let forest consisting trees txj least one vertex txj distinguished next size claim detection pair proof claim prove tree must contain detector assume contradiction tree txa detector belongs txa definition txa contains vertex distinguished hence vertex say txb separated listener none dominated watcher must belong txa hence indeed definition two listeners whose path contains clearly would separate contradiction since belong txa listener outside txa separate clearly watcher outside txa dominate proves claim claim discard combination hence assume tree txj build subtree txj follows remove legs txj whose closest special branching point one leg attached remove leaf txj adjacent watcher also denote forest containing tree txa next claim following claim detection pair detectors txa leaves proof claim let consider legs containing detector construction txa leg must attached special branching point least two legs attached note operation create new special branching point moreover leg length unique vertex dominated watcher since set detectors leg special branching point determined combination note create new leg leg contains detector watcher let special branching point let set legs txa attached contain detector previous discussion leg length observed watcher hence order separate neighbours legs need detector least legs otherwise still need many detectors order separate vertices distance legs hence half leaves belong leg detector proves claim claim txj leaves set detectors placed txj therefore total number leaves forest discard current combination hence assume total number leaves therefore total number vertices degree least let consider sets maximal threads paths degree endpoints threads vertices degree least therefore number threads number vertices thread puv two vertices tree corresponds txj path whose either single leg attached contain watcher set leaves attached hence may assume puv contains two listeners contained two lemma could replace two endpoints puv moreover puv contained exactly one listener similarly may assume placed one endpoints puv indeed since another listener solution set listener closer similarly lemma listener would distinguish contains vertices puv vertices legs attached vertices puv puv separated puv moreover set pairs vertices outside puv note lemma choose place listener endpoint path puv thread endpoint leg attached instead placing listener instead place listener leaf leg therefore thread puv four possibilities placement listeners listener leaf corresponding leg attached single listener single listener listener therefore possibilities guess placement listeners tree txa placement obtain set listeners course consider placement assume listeners sought solution hence remains check whether obtain valid solution size adding watchers phase determining set watchers compute tree similarly using new set listeners similarly define set vertices neighbour set trees also let forest containing trees txj least one vertex distinguished know tree use watchers distinguish vertices yet distinguished tree let tree obtained txj removing leaves already dominated watcher process created least leaf neighbours watcher simply remove obtained tree satisfy hence apply lemma necessary augment maximality condition iii lemma thus set vertices may possibly contain watcher computed time linear trees hence total set possible placements remaining watchers size computable linear time therefore check log possibilities placing remaining watchers find valid detection pair return yes otherwise discard placement move next possibility clear lemmas used description algorithm algorithm correct running time preliminary phase log running time phases log log completes proof conclusion obtained algorithm pair trees perhaps algorithmic upper bound could improved ptas seems reduction show inapproximability pair constant greater one therefore remains settle exact approximation complexity pair trees second question factor log fpt algorithm detection pair improved fpt time even time bottleneck use lemma preliminary phase phase algorithm search possible subsets size sets size respectively moreover perhaps possible obtain linear terms reducing preliminary phase extension results would interest determine whether graph classes pair approximable whether detection pair fpt one natural research direction consider class planar graphs subclasses outerplanar graphs graphs graphs treewidth three classes contain trees although questions settled affirmative optdominating set dominating set see far know also remain open dimension metric dimension though algorithm exists dimension outerplanar graphs acknowledgements thank anonymous referees careful reading valuable comments wish thank bert hartnell sharing manuscript moreover acknowledge financial support programme idex bordeaux cpu references ausiello crescenzi gambosi kann protasi complexity approximation springer bailey cameron base size metric dimension invariants groups graphs bulletin london mathematical society belmonte fomin golovach ramanujan metric dimension bounded width graphs proceedings international symposium mathematical foundations computer science mfcs lncs chartrand eroh johnson oellermann resolvability graphs metric dimension graph discrete applied mathematics cockayne goodman hedetniemi linear algorithm domination number tree information processing letters diaz pottonen serna van leeuwen complexity metric dimension proceedings european symposium algorithms esa lncs dinur steurer analytical approach parallel repetition proceedings fortysixth annual acm symposium theory computing stoc downey fellows fundamentals parameterized complexity springer eppstein metric dimension parameterized max leaf number journal graph algorithms applications epstein levin woeginger weighted metric dimension graphs hard easy cases algorithmica fernau heggernes van hof meister saei computing metric dimension chain graphs information processing letters finbow hartnell young complexity monitoring network watchers listeners networks accepted foucaud mertzios naserasr parreau valicov identification locationdomination metric dimension interval permutation graphs algorithms complexity algorithmica appear garey johnson computers intractability guide theory npcompleteness freeman harary melter metric dimension graph ars combinatoria hartung exploring parameter spaces coping computational intractability phd thesis technische berlin germany https hartung nichterlein parameterized approximation hardness metric dimension proceedings ieee conference computational complexity ccc hauptmann schmied viehmann approximation complexity metric dimension problem journal discrete algorithms haynes hedetniemi slater fundamentals domination graphs marcel dekker haynes hedetniemi slater editors domination graphs advanced topics marcel dekker johnson approximation algorithms combinatorial problems journal computer system sciences khuller raghavachari rosenfeld landmarks graphs discrete applied mathematics niedermeier invitation algorithms oxford university press slater leaves trees congressus numerantium
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circulant graphs oct kevin vander meulen adam van tuyl catriona watt abstract let circulant graph let denote edge ideal ring consider problem determining ring graph must focus known families wellcovered circulant graphs form also characterize cubic circulant graphs end observation even though property preserved lexicographical products graphs true property introduction let denote finite simple graph vertex set edge set identifying vertices variables polynomial ring field associate quadratic monomial ideal hxi called edge ideal edge ideals first introduced villarreal last couple years interest determining graphs determining ring ring solely properties graphs although problem probably intractable arbitrary graphs results known families graphs chordal graphs bipartite graphs readers may also interested recent survey morey villarreal textbook herzog hibi especially chapter goal identify families circulant graphs given integer subset circulant graph graph vertices edge min see example graph figure convenience notation suppress set brackets set circulant graphs belong family cayley graphs sometimes viewed generalized cycles since complete graph also circulant graph literature circulant graphs appeared number applications related mathematics subject classification key words phrases circulant graph graph vertex decomposable shellable last updated january research first two authors supported part nserc discovery grants research third author supported nserc usra circulant graphs figure circulant graph networks codes even music part regular structure see classify families circulant graphs use fact graphs must graph maximal independent sets cardinality equivalently every maximal independent set maximum independent set see survey plummer algebraic graph edge ideal unmixed associated primes height families circulant graphs recently classified brown hoshino main results see theorems refine work brown hoshino determining circulant graphs also particular show theorem circulant also show graphs fact vertex decomposable shellable although wellcovered circulant graphs prove graphs buchsbaum see theorem also classify cubic circulant graphs see theorem paper structured follows section recall relevant background regarding graph theory simplicial complexes section classify cohenmacaulay graphs form section contains proof lemma needed prove main result section section look cubic circulant graphs classify section contains concluding comments open questions related lexicographical product graphs background definitions results simplicial complex vertex set set subsets satisfies note condition implies elements called faces maximal elements respect inclusion facets dimension face given dim dimension simplicial complex denoted dim maximum dimension faces call pure simplicial complex facets dimension let number faces dimension convention dim circulant graphs see theorem given simplicial complex associate monomial ideal polynomial ring field follows xjr xjr ideal commonly called ideal quotient ring ring say ring cohenmacaulay ring depth krull dimension length longest chain prime ideals strict inclusions depth depth length longest sequence forms regular sequence review required background reduced homology see complete details simplicial complex associate reduced chain complex vector space basis elements corresponding faces assume boundary maps given denotes omitted term ith reduced simplicial homology coefficients space ker dimensions related via reduced euler characteristic dimk find convenient use reisner criterion given face link subcomplex theorem reisner criterion let simplicial complex dim vertex deletion subcomplex following combinatorial topology property introduced provan billera circulant graphs definition let pure simplicial complex vertex decomposable simplex unique maximal facet exists vertex decomposable also refer following family simplicial complexes definition let pure simplicial complex facets shellable exists ordering following theorem summarizes number necessary sufficient conditions simplicial complexes require theorem let simplicial complex vertex set pure entries iii pdim denotes projective dimension length minimal free resolution vertex decomposable dim vertex dim vertex connected proof many results standard see theorem see theorem iii follows theorem see corollary fact shellable implies theorem proposition theorem paper interested independence complexes finite simple graphs say set vertices independent set independence complex set independent sets ind independent set set ind simplicial complex following convention resp shellable vertex decomposable ind resp shellable vertex decomposable facets ind correspond maximal independent sets vertices common let denote cardinality maximum independent set vertices graph every maximal independent set cardinality moreover direct translation definitions gives lemma circulant graphs characterization circulant graphs section classify circulant graphs form cohenmacaulay brown hoshino recently classified graphs family theorem theorem let integers brown hoshino result key ingredient main result also need one additional result independence polynomial translated statement independence polynomial graph given number independent sets cardinality take note ind translate theorem language independence complexes get following statement lemma let integers dim ind lemma characterize circulant graphs form suffices determine graphs theorem also interestingly proving subtle part proof carry part proof need following lemma whose proof postpone next section lemma fix integer let ind dimk assuming moment lemma holds arrive main result theorem let integers let following equivalent iii shellable vertex decomposable proof always iii prove iii lemma dim ind apply theorem dim ind let ind connected edges ind hand ind connected circulant graphs see let ind subscript addition adjusted modulo thus make path ind connected applying theorem shows iii complete proof show proof iii already showed ind connected dim ind theorem implies theorem consequently lemma therefore remains show remainder proof dedicated case lemma dim ind ind given ind ind hence ind ind hence ind two cases theorem implies therefore assume show ind show ind suffices ind linkind reisner criterion theorem would imply ind using fact dim ind given reduced euler characteristic know dimk ind ind connected simplicial complex dimk ind simplifying sides equation rearranging gives dimk ind dimk ind lemma dimk ind ind desired specialize theorem case recover known classification cycles corollary note also cycle corollary let circulant graphs even though still interesting algebraic structure noted theorem definition pure simplicial complex called buchsbaum field every face dim say graph buchsbaum independence complex buchsbaum note reisner criterion theorem buchsbaum classify circulant graphs form buchsbaum theorem let integers let buchsbaum proof buchsbaum ind must pure theorem theorem implies first show buchsbaum let ind since dim given wish show dim hence dim dim dim therefore suffices show dim symmetry assume without loss generality independent set containing extended maximal independent set furthermore independent set cardinality three turn implies dim suffices prove proving condition equivalent proving connected first note none vertices appear vertices adjacent hand following elements facets consequently following edges thus connected desired suppose shown proof theorem ind consists disjoint edges ind ind variable therefore buchsbaum circulant graphs proof lemma purpose section prove lemma interested finding induced octahedrons independence complex lemma fix integer let let ind associated independence complex let six distinct vertices induced simplicial complex isomorphic labeled octahedron figure induced graph graph three disjoint edges figure labeled octahedron proof suppose isomorphic octahedron figure follows edges octahedron also edges independence complex means set independent set words edges suffices show consists edges vertex vertex edge however edge consequently edge contradiction converse reverse argument three disjoint edges follows independent sets thus belong consequently faces facets complex eight faces whence octahedron come desired proof proof lemma begin first recalling facts ind theorem lemma simplicial complex pure two dimensional therefore reduced chain complex form follows chain complex dimk dimk ker circulant graphs linearly independent elements ker strategy therefore identify note subset vertices induced complex isomorphic octahedron octahedron corresponds element ker make precise suppose octahedron simplicial complex facets note face associate following element assumed indices basis element written increasing order boundary map evaluated gives ker compute lower bound dimk build list octahedrons order elements using lexicographical ordering octahedron list contains face appeared previous octahedron respect ordering associating octahedron corresponding element octahedron belong ker moreover fact octahedron face appeared previously implies octahedron written linear combination previous elements ker thus giving required number linearly independent elements lemma correspondence induced octahedrons induced subgraphs consisting three pairwise disjoint edges represent octahedron tuple correspond edges begin considering octahedrons described following list take list octahedrons add one index get new list octahedrons terms graph rotating disjoint circulant graphs edges right rotate disjoint edges equivalently add one index example disjoint edges rotated right times give octahedrons hand disjoint edges rotated times create octahedrons carry procedure end expanded list octahedrons see one collection disjoint edges rotated times two tuples disjoint edges rotated times arrive tuples constructed tuples rotated times suffices show corresponding elements ker linearly independent arranged list lexicographical order smallest largest consider face associated octahedron claim progress list face appeared previous octahedron circulant graphs particular suppose item wish show face appeared first octahedrons lexicographically ordered list suppose appears earlier list contains face face appear must contain exactly one must contain exactly one remaining two vertices must contain remaining vertex face way listed constructed octahedrons note otherwise contradicting lexicographical ordering also contradiction problem arises thus since must also appear tuple two possibilities neither tuples appear strictly respect ordering thus completing proof circulant cubic graphs brown hoshino classified circulant cubic graphs recall cubic graph graph vertex degree thus circulant cubic graph finite number connected circulant cubic graphs theorem theorem let connected circulant cubic graph isomorphic one following graphs using computer algebra system like one simply check graphs displayed figure theorem let connected circulant cubic graph isomorphic proof theorem suffices check graphs also figure connected cubic circulant graphs circulant graphs graph dim theorem iii simply need check pdim compute graphs figure inspection hand compute projective dimension using computer algebra system following table summarizes calculations pdim conclusion follows values table brown hoshino use following result extend theorem circulant cubic graphs following classification due davis domke theorem let let gcd even isomorphic copies odd isomorphic copies also use following lemma next proof lemma proposition suppose graph disjoint components theorem let cubic circulant graph let gcd proof suppose even odd also theorem theorem thus lemma conversely theorem isomorphic copies isomorphic copies cases theorem lemma imply concluding comments open questions question classifying circulant graphs probably intractable problem even weaker question determining whether circulant graph equivalently ind pure simplicial complex shown brown hoshino theorem present best probably expect identify families circulant graphs brown hoshino observed circulant graphs behave well respect lexicographical product recall construction definition given two graphs lexicographical product denoted graph vertex set two vertices adjacent either circulant graphs circulant graphs lexicographical product also circulant see theorem property also preserved respect lexicographical product see theorem let two graphs graphs consequence families circulant graphs discovered combined new circulant graphs using lexicographical product therefore natural ask lexicographical product allows build new cohenmacaulay circulant graphs known circulant graphs words replace theorem turns always case following example shows example let circulant graphs seen figure compute graphs figure inspection hand figure lexicographical products compute projective dimension using macaulay find dim pdim however dim pdim cohenmacaulay light example ask conditions allow conclude lexicographical product end concerning lemma using macaulay found dimk suggests inequality lemma actually equality wonder indeed true acknowledgements thank brydon eastman writing latex code produce circulant graphs russ woodroofe jennifer biermann useful discussions references bermond comellas hsu distributed loop computer networks survey parallel distrib comput boesch tindell circulants connectivities graph theory brown hoshino independence polynomials circulants application music discrete math brown hoshino circulant graphs discrete math davis domke graphs combin math combin comput circulant graphs herzog hibi monomial ideals graduate texts mathematics herzog hibi distributive lattices bipartite graphs alexander duality algebraic combin herzog hibi zheng chordal graphs combin theory ser grayson stillman macaulay software system research algebraic geometry http miller sturmfels combinatorial commutative algebra graduate texts mathematics springer morey villarreal edge ideals algebraic combinatorial properties progress commutative algebra combinatorics homology edited christopher francisco lee klingler sean janet vassilev gruyter plummer graphs survey quaestiones math provan billera decompositions simplicial complexes related diameters convex polyhedra math oper res sachkov tarakanov combinatorics nonnegative matrices translations mathematical monographs american mathematical society providence stanley combinatorics commutative algebra second edition boston topp volkmann products graphs ars combin villarreal graphs manuscripta math villarreal monomial algebras monographs textbooks pure applied mathematics marcel dekker new york department mathematics redeemer university college ancaster canada address kvanderm department mathematical sciences lakehead university thunder bay canada address avantuyl department mathematics redeemer university college ancaster canada address cwatt
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jan python abstractions optimal checkpointing inversion problems navjot jan michael lange imperial college london london imperial college london london imperial college london london mathias louboutin andrea walther simon funke university british columbia vancouver canada paderborn paderborn germany simula research laboratory lysaker norway gerard gorman imperial college london london abstract inversion optimization problems often rely solving adjoint problem calculate gradient objective function requires storing large amounts intermediate data setting limit largest problem might solved given amount memory available checkpointing approach reduce amount memory required redoing parts computation instead storing intermediate results revolve checkpointing algorithm offers optimal schedule trades computational cost smaller memory footprints integrating revolve modern python hpc code combining code generation straightforward present api makes checkpointing accessible code generation environment along initial performance figures focus seismic applications seismic inversion computationally intensive technique uses data seismic wave propagation experiments estimate physical parameters earth subsurface seismic inversion problem based wave equation viewed optimization problem numerically solved using optimization since gradient usually calculated using method method requires forward adjoint field known time step simulation discuss section previous work similar inverse problems led revolve algorithm associated tool provides optimal schedule store checkpoints states forward simulation restored study optimal checkpointing seismic inversion done accompanied abstraction made integration software easier revolve algorithm discussed section revolve tool algorithm however provide schedule used checkpointing although eases complexity application code using algorithm glue code required manage forward adjoint runs still quite complex acts deterrent widespread use algorithm community paper describe revolve algorithm combined code generation make checkpointing much accessible software enable described section although use particular examples seismic imaging abstraction software proposed quite general nature used problem requires checkpointing combination variety computational methods section provide initial performance figures judged correctness performance implementation ccs concepts software engineering software design engineering keywords hpc code generation api checkpointing adjoint inverse problems acm reference format navjot kukreja jan michael lange mathias louboutin andrea walther simon funke gerard gorman python abstractions optimal checkpointing inversion problems proceedings supercomputing denver colorado usa november pages https corresponding author email nkukreja 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 november denver colorado usa copyright held acm isbn https introduction seismic imaging devito seismic imaging techniques exploit principle traveling wave carries information physical properties medium travels different techniques focus november denver colorado usa kukreja dsyn dobs data residual jacobian forward operator forward wavefield time derivative adjoint wavefield seen evaluation gradient first requires simulation forward adjoint wavefields achieved modeling wave equation using discretization usually finite difference various forms wave equation exist acoustic isotropic anisotropic elastic models physics different levels corresponding levels complexity focus acoustic equation although analysis applies forms equation mentioned discrete form acoustic wave equation written following linear system pts figure graphical demonstration seismic experiment produces data used input seismic imaging workflow source open university different kinds information objectives focus migration rtm imaging method relies good estimate velocity model obtain image reflectors subsurface algorithm relies datafitting procedure synthetic data dsyn computed current estimate physical model via solve compared field measured data dobs example field data recording illustrated figure problem leastsquare minimization introduce formulation problem solved justify implementation optimal checkpointing previously introduced devito domain specific language dsl pde solvers devito provides symbolic abstractions define forward adjoint wavefields implementation concentrate computation image subsurface seismic imaging case migration rtm provides image subsurface reflectors field recorded data cinematically correct smooth background velocity model practice recording repeated different pair called experiments physical region estimate physical parameters obtained recorded data different methods inversion fwi iterative rtm low frequencies estimated rtm provides image subsurface interpreted stated rtm single gradient fwi objective written minimize dsyn dobs square slowness model physical property medium wave propagating gradient objective function respect square slowness given discretized sourcerestriction operator wavefield given pts although equation provide value entire domain every time step explicitly formulating entire matrix prohibitively expensive terms computer memory required avoided wherever possible simulations interest value certain predetermined locations simulated domain call receivers record progression locations time represented mathematically applying restriction operator required receiver locations result applying call simulated data given dsyn pts rewrite objective function equation pts dobs equation define jacobian forward operator minimize dpr pts term equation calculated equation term calculated adjoint equation given ptr implementation forward wavefield obtained procedure forward time adjoint wavefield obtained similarly backward time procedure procedure derive image rtm summarized compute synthetic data dsyn forward solve equation compute adjoint wavefield data residual equation compute gradient correlation forward adjoint wavefield equation first step know equation modeled devito using operator defined figure python abstractions optimal checkpointing inversion problems def forward model eta src rec create wavefield function timedata derive stencil symbolic equation eqn eta stencil solve eqn stencil add source injection receiver interpolation create operator source receiver terms return operator november denver colorado usa def gradient model eta src rec create adjoint wavefield function timedata grad grad adjoint equation eqn eta stencil solve eqn eqn stencil add expression receiver injection receivers return operator eqn receivers figure devito code required forward operator operator thus created used model forward contains source injected field extracting receiver information simulation clearly third step requires intermediate data previous steps storing forward adjoint wavefields memory would naive since two complete wavefields need stored first obvious optimization merge steps single pass step gradient calculation use output step adjoint wavefield calculated hence saving need storing adjoint wavefield memory operator combined steps created devito using code given figure still leaves requirement result step available efficient computational point view would store full history forward wavefield first step however realistically sized models would require terabytes direct access memory one solution would store field disk would lead slow access memory usage making inefficient memory limit leads checkpointing storing subset time history recomputing adjoint propagation revolve provides optimal schedule checkpointing store given model size number time steps available memory next section discusses checkpointing implemented revolve seen usage adjoint methods allows computation gradient information within time small multiple time needed evaluate underlying function however nonlinear processes like one saw previous section memory requirement compute adjoint information principle proportional operation count figure devito code required operator calculates adjoint gradient single pass underlying function see sec chap book several checkpointing alternatives reduce high memory complexity discussed checkpointing strategies use small number memory units checkpoints store system state distinct times subsequently recomputation information needed adjoint computation available performed using checkpoints appropriate way several checkpointing techniques developed seek acceptable compromise memory requirement runtime increase obvious question place checkpoints forward integration minimize overall amount required recomputations develop corresponding optimal checkpointing strategies one take account specific setting application fixed number time steps perform constant computational cost time steps calculate simplest situation shown case checkpointing scheme based binomial coefficients yields given number checkpoints minimal number time steps recomputed obvious extension approach would include flexibility respect computational cost time steps example one uses implicit time stepping method based solution nonlinear system number iterations needed solve nonlinear system may vary time step time step yielding time step costs situation longer possible derive optimal checkpointing strategy beforehand heuristics developed tackle situation however extensive testing showed even case nonuniform step costs binomial checkpointing quite competitive another important extension coverage adaptive time stepping november denver colorado usa revolve steps snaps whatodo revolve switch whatodo case advance oldcapo capo forward case firsturn init adjoint case youturn adjoint case takeshot store xstore check case restore restore xstore check whatodo terminate figure revolve algorithm calls application interface case number time steps performed known beforehand therefore online checkpointing strategies developed see finally one take account checkpoints stored checkpoints stored memory lost failure sake resilience future supercomputers may memory constrained checkpoints may necessarily stored disk therefore access time read write checkpoint negligible contrast assumption frequently made development checkpointing approaches contributions extend available checkpointing techniques hierarchical checkpointing see software revolve implements binomial checkpointing online checkpointing described hierarchical also called checkpointing derived purpose provides data structure steer checkpointing process storage information required several checkpointing strategies illustrate principle structure adjoint computation using checkpointing fig illustrates kernel revolve used binomial checkpointing two remaining checkpointing strategies implemented similar fashion taking additional extensions account forward integration well corresponding adjoint computation performed within structure fig steps snaps denote number time steps forward simulation number checkpoints available adjoint computation respectively hence routine revolve determines next action performed must supported application differentiated actions advance user supposed perform part forward integration based routine forward represents state system control variable oldcapo contains current number state forward integration starting holds state time oldcapo variable kukreja oldcapo determines targeted number state forward integration therefore capo oldcapo time steps perform propagate state time oldcapo time capo firstrun action signals start adjoint computation therefore first target function evaluated user possibility initialize adjoint variable subsequently first adjoint step performed youturn next adjoint step performed takeshot user supposed store current state checkpoint number check array checkpoints denoted xstore specific organisation checkpoints completely user adjoint computation check selects checkpoint number appropriately states needed adjoint computation available adjoint computation started states stored checkpoints also overwritten reuse memory restore content checkpoint number check copied state recompute forward integration starting state important note checkpointing approach completely independent method actually used provide adjoint information seen adjoint computation available implementation incorporate binomial checkpointing reduce memory requirement also stress revolve provides serial checkpointing means one forward time step one adjoint step performed stage adjoint computation nevertheless computation forward time step adjoint step may performed heavily parallel may evaluated large scale computer system contrast parallel checkpointing techniques several forward time steps might performed parallel even conjunction one adjoint step corresponding optimal parallel checkpointing schedules developed however far implementation steer parallel checkpointing process available revolve software also includes adjust procedure computes given number time steps number checkpoints increase spatial complexity equals approximately increase temporal complexity using computed number number checkpoints minimises cost assuming user pays computational resources per node per time cost proportional available memory runtime computation abstractions checkpointing section discuss package pyrevolve developed course work encapsulate revolve checkpointing high level python library library available online along source first provide details implementation section afterwards discuss interplay pyrevolve checkpointing implementation previously discussed section finally discuss usage pyrevolve application https python abstractions optimal checkpointing inversion problems section rtm described section implemented devito domain specific language although section discusses special case devito interface pyrevolve library designed allow easy integration python codes well november denver colorado usa revolve original crevolve api pyrevolve side pyrevolve interface designed part providing checkpointing users devito accessible api design following goals making checkpointing available users devito without forcing get involved implementation details like loops callbacks data storage mechanisms user choose whether use checkpointing one place forced anything beyond knowledge checkpointing different strategies checkpointing multistage shall contained within one module python framework still benefiting operations code checkpointing contained separate library allows others use easily even interested using devito matured pyrevolve since data movement requires intricate knowledge data structures used organization memory handled application code achieve goals pyrevolve designed following overall workflow explained detail following sections term application refers application using pyrevolve library case devito begin application creates objects apply method perform actual forward reverse computations instances concrete implementation abstract base classoperator application also creates instance concrete implementation abstract base class checkpoint working data operators require specified memory location next application instantiates pyrevolve revolver object passes forward reverse operators checkpoint object required application starts revolver forward sweep complete forward computation store checkpoints necessary forward sweep completes application finalize computation based forward data evaluation objective functions store final result necessary may initialization adjoint data structures application calls revolver reverse sweep compute adjoint possibly performing partial forward sweeps loading checkpoint data pyrevolve package contains crevolve thin wrapper around previously published files package slightly modified compatibility python original available link footnote crevolve wrapper around library taken http https wrapper pyrevolve python wrapper devito application figure packages overview devito example application uses checkpointing subject section revolve described section packages pyrevolve crevolve used create abstraction checkpointing explained section one key design aspect pyrevolve responsible performing data copies therefore need know properties structure data needs stored purpose pyrevolve provides checkpoint abstract base class size attribute load ptr save ptr method user must provide concrete implementation object size attribute must contain size single checkpoint memory information used pyrevolve allocate correct amount memory save method must working data memory region starting provided pointer load method must restore working data memory region starting pointer ptr either performing pointing computation existing data inside checkpoint storage pyrevolve library provides user class revolver must instantiated following arguments checkpoint object implementation abstract base class checkpoint forward operator object provides function apply specified section performs forward computation reverse operator similarly object provides function apply performs reverse computation number checkpoints optional specifies number checkpoints stored memory given default value computed using adjust method explained section november denver colorado usa kukreja action crevolve advance takeshot restore firstrun youturn error capo uint oldcapo uint check uint action figure crevolve classes example field needs checkpointing timedata expression generates values operator expression uses values rev operator devitocheckpoint revolver revolver rev forward sweep pause take checkpoints could perform additional steps reverse sweep uses checkpoints checkpoint size int figure devito code utilize checkpointing based pyrevolve save ptr pointer void load ptr pointer void storage data init int int uint pointer revolver operator operator store storage crevolve checkpointer ckp checkpoint init ckp void void operator apply void figure pyrevolve classes abstract classes checkpoint operator implemented client application number time steps also optional given online checkpointing algorithm used based either given computed number checkpoints constructor instantiates storage object allocates necessary amount memory number checkpoints checkpointsize makes memory accessible revolver api application side introduce checkpointing using pyrevolve library application must implement particular interface use devito example application however everything discussed fairly general application may implement checkpointing using pyrevolve using approach discussed section begin concrete implementation pyrevolve abstract base class checkpoint called devitocheckpoint created class three methods save ptr save contents working memory location ptr restore ptr restore previously stored checkpoint location ptr working memory size report amount memory required single checkpoint used decide total amount memory allocated calculate offsets along set two operator one forwardoperator carry forward computation gradientoperator computes image explained section used initialize revolver object shown figure initialization revolver object devitocheckpoint object queried size one checkpoint pyrevolve allocates bytes memory storage checkpoints calling carries forward run broken chunks specified checkpointing schedule provided revolve chunk executed calling arguments corresponding timesteps run simulation chunks automatically called save state checkpoint calling revolver calls relevant arguments sections result forward pass available checkpoint loaded call others automatically call recompute store memory results part forward operator reverse operator applied part python abstractions optimal checkpointing inversion problems november denver colorado usa devitocheckpoint size int save ptr pointer void load ptr pointer void forwardoperator reverseoperator data data apply void apply void figure velocity model used gradient test figure devito operators implementation checkpoint class following work users devito easily add optimal checkpointing adjoint computations following steps described experiment two possible ways testing numerical accuracy implementation solving problem whose solution certain known mathematical properties verifying properties numerically comparing results reference solution use gradient test described also verify numerical results match reference implementation test uses taylor property gradient test whether calculated gradient follows expected convergence small perturbations test written mathematically figure timings gradient test different amounts peak memory consumption hdm hdm defined equation test carried certain call smoothed version two layer model true model two horizontal sections different value squared slowness figure shows true velocity model measured data required objective function modeled true model constant varies verify first order error second order error code used test found repository devito tests carried intel xeon cpu haswell ram used grid points simulation running timesteps required store full forward wavefield memory first run made regular gradient example stores entire forward wavefield memory repeated checkpointing varying number checkpoints verified results https versions matched exactly also passed gradient test mentioned previously peak memory usage tracked run well total time solution memory consumption measured using python module time solution using time python command function profiled eliminate variation results timings minimum value three runs seen figure reduction runtime memory available line theoretical predictions griewank walther expected theoretical numbers take account cost deep copies implemented using numpy well cost repeatedly calling function python repetition inside function since adjoint computation well associated forward computation carried one time significantly reduces amount work available inside single operator call might cause inefficiencies across cores using openmp effect seen clearly comparing reference implementation stores forward november denver colorado usa field contiguous block memory checkpointed implementation stores checkpoint every time step case although memory consumption two implementations checkpointed implementation runs slower overheads previously mentioned conclusions work shown abstractions possible greatly simplify complexity client code also verified correctness implementation using mathematical tests already enables users devito utilize revolve based checkpointing applications solve much bigger problems previously possible however experiment section seen overhead introduced checkpointing reason much work done implement features widen applicability pyrevolve future work work carried far proof concept integration revolve using high level abstractions still terms use practical applications important limitation current implementation implements serial checkpointing reverse computation one timestep advanced time severely limits parallelizability code interface would need extended able manage parallelization strategies another important feature might required pyrevolve adopted community checkpointing checkpoints may transparently swapped disk increasing amount memory available applications without change application code problems implemented adaptive number known would require pyrevolve implement online checkpointing something even devito would require future versions acknowledgments authors grateful fabio luporini nicolas barral work carried part intel parallel computing centre imperial college london references august http aupy herrmann hovland robert optimal algorithm adjoint computation siam journal scientific computing edip baysal dan kosloff john sherwood reverse time migration geophysics https arxiv http andreas griewank andrea walther algorithm revolve implementation checkpointing reverse adjoint mode computational differentiation acm transactions mathematical software toms griewank walther evaluating derivatives principles techniques algorithmic differentiation siam eldad haber matthias chung felix herrmann effective method parameter estimation pde constraints multiple right hand sides siam journal optimization http kukreja navjot kukreja mathias louboutin felippe vieira fabio luporini michael lange gerard gorman devito automated fast finite difference computation languages frameworks high performance computing wolfhpc sixth international workshop ieee michael lange navjot kukreja fabio luporini mathias louboutin charles yount jan gerard gorman optimised finite difference computation symbolic equations arxiv preprint plessix review method computing gradient functional geophysical applications geophysical journal international schanen marin zhang amd anitescu asynchronous twolevel checkpointing scheme adjoints solver technical report argonne national laboratory julia sternberg michael hinze implementation method optimal control nonlinear pdes optimization methods software stumm walther approaches optimal offline checkpointing siam journal scientific computing philipp stumm andrea walther new algorithms optimal online checkpointing siam journal scientific computing william symes reverse time migration optimal checkpointing geophysics https arxiv https albert tarantola inversion seismic reflection data acoustic approximation geophysics https arxiv https jean virieux operto overview inversion exploration geophysics geophysics andrea walther bounding number processes checkpoints needed parallel reversal schedules computing qiqi wang parviz moin gianluca iaccarino minimal repetition dynamic checkpointing algorithm unsteady adjoint calculation siam journal scientific computing https
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fast parameterized algorithm set blair sullivan andrew van der poel department computer science north carolina state university raleigh jul blair sullivan january abstract set problem asks given graph positive integer whether one delete edges remainder collection disjoint paths give fpt algorithm complexity deciding set significantly improving previously best known feng zhou wang main tool new algorithm set using cut count framework denotes treewidth general graphs combine branching algorithm refines reduced instances prove bounded treewidth introduction study parameterized versions set problem asking minimum number edges whose deletion graph results collection disjoint paths deleted edges set see figure specifically concerned set uses natural parameter number edges deleted set input graph integer parameter problem exist size exactly set disjoint paths problems naturally motivated determining ordering genetic markers dna using fragment data created breaking chromosomes gamma radiation technique known radiation hybrid mapping unfortunately human error distinguishing markers often means constraints implied markers fragments incompatible possible linear orderings necessitating algorithm find best ordering violates fewest constraints set solves special case dna fragment contains exactly two genetic markers corresponding edge graph linear ordering markers must correspond figure three sets dashed edges including one minimum size rightmost set paths minimize number unsatisfied constraints edges set recent algorithmic results related set include algorithm two parameterized algorithms deciding set faster time however written parameterized results contain flaw analysis invalidates probability correct solution given best known bound prior algorithm paper prove theorem set decidable time probability least note standard amplification arguments apply theorem holds success probability less increasing function solve set success probability least npolylog remainder paper organized follows essential definitions notation section start section giving new algorithm solving set parameterized treewidth using cut count framework finally section describes algorithm referenced theorem solves set general graphs applying set reduced instances generated via kernelization branching preliminaries let graph vertex set edge set unless otherwise noted assume let denote set neighbors vertex let deg given graph write graph algorithm section uses dynamic programming tree decomposition running time depends related measure treewidth denote simplify dynamic programming throughout paper use notation fpt tractable complexity say algorithm complexity step versions algorithm checks candidate set size sweeping possible sizes candidates want restrict size accordingly large algorithm discards continues next iteration however order analysis hold probability candidate contained set must every iteration candidates large may significantly smaller probability containment yet counted exponent analysis properties reduced instances guarantee find tree decomposition poly time use variant nice tree decompositions node tree one five specific types leaf introduce vertex introduce edge forget vertex join introduce edge nodes labelled edge one child identical bag require edge introduced exactly additionally enforce root node type forget vertex thus empty bag tree decomposition transformed nice decomposition width time linear size input graph describing dynamic programming portion algorithm use iverson bracket notation predicate let jpk true otherwise also use shorthand denote updating function values unchanged finally use fast subset convolution reduce complexity handling join nodes nice tree decomposition section technique maps functions vertices join bag values chosen based application key complexity result rely uses product defined write set vectors length assigning value element definition product let fixed integer let finite set say ring functions define product fast subset convolution guarantees certain products computed quickly lemma cygan let constant product functions computed time ring operations algorithm via cut count start giving fpt algorithm set parameterized treewidth primary tool cut count framework enables ctw onesided monte carlo algorithms problems constant probability false negative cut count previously used improve bounds several problems including connected vertex cover hamiltonian cycle feedback vertex set pilipczuk showed ctw algorithm constant designed cut count approach set problem expressed specialized graph logic known however since end goal improve existing algorithms set general graphs using bounded treewidth kernel need develop specialized dynamic programming algorithm small value show theorem exists fpt monte carlo algorithm deciding set graph time failure probability cut count technique two main ingredients algebraic approach counting uses arithmetic enabling faster algorithms alongside guarantee undesirable objects seen even number times result implies desired solution seen idea defining problem connectivity requirement consistent cuts context consistent cut partitioning vertices graph two sets edge vertices degree since connected component must lie completely one side consistent cut see graph exactly cuts number connected components number isolated vertices order utilize parity number consistent cuts introduce markers create even numbers consistent cuts graphs collections disjoint paths counting algorithm computes parity size collection subgraphs consistent cuts adhere specific properties pertaining set employs dynamic programming nice tree decomposition use weights isolation lemma bound probability false negative arising multiple valid markings solution use fast subset convolution reduce complexity required handling join bags dynamic programming remainder section present specifics applying techniques solve set cutting first provide formal definitions markers marked consistent cuts use ensure sets disjoint paths counted exactly entry dynamic programming table definition triple marked consistent cut graph consistent cut refer edges markers marker set proper contains least one edge connected component note marker set proper vertices side cut definition consistent cut isolates side every connected component contains marker connected components must fall entirely side well therefore proper marker set exists exactly one consistent cut marker sets proper paired even number consistent cuts unmarked components may lie use proper marker sets distinguish desired subgraphs assigning markers way prune dynamic programming table solutions described later section subgraphs consider may proper marker set collections disjoint paths know marker set proper subgraph unique consistent cut thus collections disjoint paths counted entry dynamic programming table subgraphs counted even number times note claiming collections disjoint paths proper marker sets refer complement set edges disjoint paths call paired proper marker set size exactly equal number connected components viewed solutions due complementary nature marked crucial counting algorithm thus subgraphs correspond solutions dynamic programming table describe use isolation lemma guarantees able use parity distinguish solutions let denote isolation lemma let set family universe function said isolate unique minf assign weights uniformly random probability isolates least intuitively set solutions complements solutions instance set even would return false negative solution counted odd number times even number solutions total count solutions even making combined count solutions even algorithm would incorrectly determine solution exist false negative isolation lemma allows partition based weight solution assigned guarantees least one partition blocks odd size constant probability let contain two copies every edge one representing marker one edge denotes pairs edge subsets potential set selected achieve success probability theorem copy edge assigned weight uniformly random probability finding isolating thus denote values assigned set marker copies likewise set copies counting graph corresponds set size number edges markers match specific values depend values easily deduced know deletion set solution size leave edges furthermore forest connected components number markers needs isolates forest removed resulting graph still forest thus actual number markers necessary describe dynamic programming algorithm nice tree decomposition returns mod number appropriately sized root subtree fixed since appropriately sized least one odd parity number size corresponding implies solution set instance must exist algorithm actually count values number subgraphs maximum degree exactly edges marked consistent cut markers following lemma justifies counting place lemma parity number edges weight parity number edges markers weight proof consider subgraph maximum degree edges let marking assume first collection paths know connected components proper marker set exactly one consistent cut therefore contributes one number number respectively otherwise proper marker set contains unmarked connected component even number consistent cuts therefore contributes even number count zero number finally contains least one cycle therefore least one connected component contain marker number consistent cuts even contribution count even contribution count zero conclude parity number parity number dynamic programming algorithm approach nice tree decomposition build values encoding option add edges select edges markers keep track various parameters ensuring pruning table consider could valid solutions set instance use number edges ensure solution correct size number markers vertices determine subgraph acyclic weight parameter allows distinguish solutions decreases likelihood false negative occurring via isolation lemma finally need parameter encodes degree information required properly combine iterate tree call parameter define vertices bag variable parameter maximum value vertices edges markers weight edges markers table dynamic programming table parameters upper bounds corresponds degree associated table entry vertices degree value denotes side partition vertices degree side cut definition need keep track side cut similarly degree vertices additional incident edges thus side cut fall matter selecting markers summary table entries counting number bag vertices edges markers weight following description dynamic programming algorithm nice tree decomposition let denote children join node otherwise unique child denoted leaf inputs introduce vertex introduce edge kay subs function returns values degreefunction child node could assigned vertices based current summarized subs argue formula correctness term handles case excluded handle case added iterating possible subs values endpoint considering counts child entries appropriate subs values preventing ever vertex degree greater note use function guarantee labels isolate use introduced edge summation possible values order consider falling either side cut formulation assures endpoint degree included count degree utilize marker weight distinguish choose marker side cut increment accordingly either case update appropriately marker marker introduced forget vertex forgotten vertex degree must consider possible values assigns child bag note cccandidates isolate member connected component contains marker cancel mod either side cut parameters identical join compute via fast subset convolution taking care combine table entries whose compatible definition join node children compatible one following holds every vertex order apply lemma let bag transform values assigned degree function values let defined table extending vectors application use apply lemma function corresponds vertex degree used tandem ensure compatibility requirements met necessarily table easy verify together imply compatible sum functions computing values join nodes make sure solutions children combined compatibility assign accordance lemma let sum degrees vertices join node assigned defining functions follows writing vector hdi compute point exactly compatible conclude section describing search table root node lemma parity number edges weight parity number edges markers weight candidates recorded table entries number therefore exists set note entry vertices contained root node definition lemma time complexity join node note four types bags consider one instance per table entry complexity point size table polynomial linear number bags polynomial number entries combinations parameters bag since nice tree decomposition size linear bottomup dynamic programming runs total time complexity bound combined correctness discussed proves theorem achieving general graphs order use solve set graphs unbounded treewidth combine kernelization branching procedure generate set reduced instances bounded treewidth subgraphs input graph specifically begin constructing kernel size described reduced instances bounded degree subgraphs kernel given branching technique prove least one reduced instance equivalent instance bound number reduced instances reduced instance bounded treewidth finally analyze overall computational complexity process kernelization branching start describing branching procedure algorithm uses technique similar zhang implementation takes instance set two integers returns set reduced instances subgraph exactly edges maximum degree least one equivalent instance size output hence running time depends input parameters select achieve desired complexity copath section also make use budget parameter keeps track many edges removed per constraints initially set branching procedure leverages observation set exists every vertex two incident edges specifically every vertex degree greater branch pairs incident edges could remain removing valid set calling pair candidate creating search tree subgraphs algorithm returns set reduced instances size number leaves search tree branching process inequality result algorithm discarding branches number edits necessary branch vertex exceeds number allowed deletions remaining give upper bound size set algorithm generating reduced instances algorithm let vertex maximum degree deg select vertices uniformly random return else deg return else return discard lemma let search tree formed number leaves proof lemma number children interior node depth resulting leaves depth limited second condition line algorithm recursive call decremented initally set implies depth proving claim finally argue least one member set reduced instances returned equivalent original consider solution set original instance every vertex two incident edges since candidates considered every vertex least one branch correctly keeps edges treewidth reduced instances algorithm produces reduced instances bounded degree order bound treewidth make use following result originated lemma extended lemma exists every graph vertices number vertices degree number vertices degree least given table moreover tree decomposition corresponding width constructed polynomial time since structure set naturally provides constraints degree sequence able apply lemma reduced instances effectively bound treewidth first find upper bound number vertices set table numerically obtained constants used lemma originally given table lemma let number vertices degree graph maximum degree set proof since removing set edges results graph maximum degree vertex degree least incident edges must removed thus removed edge counts twice endpoint lemma let instance set vertices max degree treewidth bounded constant tree decomposition corresponding width constructed polynomial time proof let defined lemma let graph formed adding isolates lemma maximum degree substitute bound lemma yields note inequality holds pair negative terms corresponding terms value since constant since treewidth monotone subgraph inclusion proves claim point applying lemma reduced instances computing desired tree decomposition polynomial since subgraphs algorithm copath section describes combine techniques prove theorem shown algorithm start applying find kernel size process deletes edges guess number edges remove branching use create set reduced instances edges note considers possible reduced instances thus solution exists contained least one reduced instance ensure complexity finding reduced instances dominate running time set degree bound reduced instances choice valid considering possible values assured contains reduced instance passed correctly decides problem probability algorithm deciding set algorithm copath foreach return true return false proof theorem analyze running time copath given algorithm lemma size reduced instance lemma applying theorem runs time reduced instance success probability least iteration outer loop completed time use choose since loop runs times also bound overall computational complexity copath additionally copath kernelization kernel size avoiding additional poly complexity subroutine note lemma tree decomposition found polynomial time size reduced instance since reduced instances subsets linearity unaffected graph size polynomial conclusion paper gives fpt algorithm set coupling kernelization branching derive algorithm deciding significantly improving previous result believe idea combining branching algorithm guarantees equivalent instances bounds degree sequence problem constraints applied problems order obtain bound treewidth allowing approaches extended general graphs one natural question whether similar techniques extend generalization set hypergraphs treated zhang also open whether combined parameterization asking set size resulting disjoint paths solvable fpt time acknowledgements work supported part gordon betty moore foundation ddd investigator award darpa graphs program spawar grant opinions findings conclusions recommendations expressed publication author necessarily reflect views darpa ssc pacific moore foundation thank two anonymous reviewers providing simplification previous branching algorithm pointing result enabling branch vertices degree greater also thank felix reidl helpful suggestions earlier draft significantly improved presentation results references husfeldt kaski koivisto fourier meets fast subset convolution proceedings stoc pages chen lin wang approximation algorithm minimum set problem algorithmica cheng cai goebel lin zhu radiation hybrid map construction problem recognition hardness approximation algorithms unpublished manuscript cox burmeister price kim myers radiation hybrid mapping somatic cell genetic method constructing maps mammalian chromosomes science cygan nederlof pilipczuk pilipczuk van rooij wojtaszczyk solving connectivity problems parameterized treewidth single exponential time focs pages ieee feng zhou randomized parameterized algorithms set problem faw pages springer feng zhou wang kernelization randomized parameterized algorithms set problem comb press doi fomin gaspers saurabh stepanov two techniques combining branching treewidth algorithmica april gaspers exponential time algorithms structures measures bounds vdm kloks treewidth computations approximations volume lncs springer mulmuley vazirani vazirani matching easy matrix inversion proceedings stoc pages acm pilipczuk solving connectivity problems parameterized treewidth single exponential time mfcs pages richard iii withers meeker maurer evans myers cox radiation hybrid map proximal long arm human chromosome containing multiple endocrine neoplasia type disease loci hum robertson seymour graph minors algorithmic aspects algorithms slonim kruglyak stein lander building human genome maps radiation hybrids comp zhang jiang zhu radiation hybrid map construction problem parameterized journal combinatorial optimization
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approach array content analysis aug graeme jorge peter harald peter department computing information systems university melbourne victoria australia gkgange schachte harald pstuckey nasa ames research center moffett field usa abstract present parametric abstract domain array content analysis method maintains invariants contiguous regions array similar methods gopan reps sagiv halbwachs however introduces novel concept array content graph avoiding need factorial partitioning step resulting analysis used arbitrary numeric relational abstract domains evaluate domain range array manipulating program fragments introduction imperative programming languages offer mutable arrays however owing indirect relation storage retrieval arrays particularly amenable static analysis analysis array bounds well studied recently real progress analyzing array content early approaches involved array smashing entire array treated single symbolic variable used concept weak updates transfer functions array assignment weaken previous abstract state weak updates generally lead rapid loss precision different segments array different properties significant improvement gopan reps sagiv use array partitioning split arrays symbolic intervals segments partitioning facilitated initial analysis array index expressions determine relative order indices denote key idea distinguish segments represent single array cells represent multiple cells permits strong updates singleton segments array partitioning method selects small set partition variables maintaining disjunctive information properties hold feasible total orderings partition variables halbwachs extended approach support relational content dox mains limited form quantified inwhile variants resulting method precise drawbacks first reif quires initial segmentation phase set partition variables identified phase purely syntactic else possible variables omitted critical invariant second exponentially many possible total derings partition variables many else partition variables identified analyarim sis may become prohibitively expensive example init randm family programs shown figure number partitions loop head follows fig init randm family sion increases program fragments discussed detail appendix finally analysis support arbitrary manipulation index variables indices may incremented decremented cousot instead maintain single partitioning array selecting consistent totally ordered subset scalar variable analysis require separate segmentation phase saving considerable overhead however considers single consistent ordering supports value domains invariants derives quite weak consider init fixed element assigned either case relationship known must select either either case desired invariant loop exit lost alternative approach expressing array properties lift abstract domain quantified invariants technique quite general two major limitations first requires user specification templates describe quantifiers introduced second expensive owing computation example join formulas must compute since negative positions prohibitively expensive many domains dillig replace strong weak updates fluid updates method value analysis relational sense builds graph nodes represent abstract locations include arrays qualified index variables edges represent constraints index variables identify concrete elements source location point concrete location target fluid update removes dichotomy strong weak updates computing first constraint representing elements modified update adds new edge strong update adding negation existing edges source weak update negation thus would unsound add directly edges instead analysis produces bracketing constraints pairs underapproximations negation done sound manner analysis expressive avoiding large number explicit partitions fixed priori however method still expensive since whenever array accessed edges must modified adding possibly disjunctive formulas propose new approach array content analysis extend existing scalar domain introducing refer segments array selecting index expressions nodes graph annotating graph edges properties hold segments arrays index expressions array content graphs offer greater flexibility approaches allow reason properties hold contiguous array segments without committing single total ordering index expressions still taking advantage available partial ordering information result array content analysis fully automatic used arbitrary domains incur factorial cost previous methods particular used relational analyses accounts possibility array elements related array indices base presentation small control flow graph language instructions array assignments jumps blocks programs constant label label label error end label basic block possibly empty sequence instructions ending possibly conditional jump arithmetic unary binary operators denoted respectively comparison operators assume fixed set arrays global scope overlap memory semantics conventional discussed figure shows example program diagrammatic form analysis assumes abstract domain analysis fragment language obtained leaving scalar analysis use parametric domain construct array content analysis remainder paper structured follows section introduces method underlying ideas section discusses computational details efficiency method evaluated experimentally section gives report section concludes suggesting work array content domain let sets scalar array variables respectively state concrete domain pair maps scalar variables integer values maps array cells let denote set variables set constants may act segment bounds use extended set variables head represents value guard allows talk properties apply single array cells constant vertices often needed array properguard ties often hold ranges bounded conif body tail stant one side array processing code often initializes fixed set indices body processing rest array suffitail aris cient consider variables conend bris stants used directly indices consider copy program shown figure case guard neither ever used directly index define boundary conditions needed invariant interest fig copying array practice however often considerably smaller wish relate value elements array segment variables scalar domain state abstract domain form expresses scalar properties expresses array content properties array allocate corresponding variable segment variable occurs never relations scalar segment variables captured use denote set segment variables sometimes may wish relate values array segment corresponding index example prove aris across segment support introduce variable idx represent index given read use tidxu denote augmented set segment variables analyses scalar domain array contents based lattice represent program state pair scalar properties matrix array properties denotes properties hold indices interval slight abuse notation use denote formula symbolic array variable replaced corresponding array element simplicity assume arrays elements integers extension arbitrary types may include integers difficult complexity array elements acting indices present consider fig array content graph assignment aris figure vertices matrix entries corresponding omitted reading edge given numeric abstract domain set arrays scalar variables array content domain pair value matrix assume function evala constructs new state treats element array scalar variable evala concretization function defined components terms concretization function scalar domain evala jpv notice constraints first equation array variable assignment type value content domain cartesian product fixed set elements also forms lattice possesses corresponding fixed point properties edge assume corresponding interval rri jss note edge constraint since always true since interval clearly empty given state representation take join two abstract states piecewise application join compute meet analogously see analysis works consider program figure figure shows abstract state immediately executing aris array content information given matrix array properties array content graph shown upper right really way visualizing matrix note infeasible edges labelled omitted fact shall usually show transitive reduction array content graph edge whose value given omitted depict edges representing without label leads shorthand graph figure lower right normalization abstract states given set constraints form wish normalize state computing strongest consequences still form critical observation property holds must also hold range compute strongest consequences must compute greatest fixed point rewrite system derived set inequalities tempting try compute fixed point using obhead vious rewrite system guard shall see formulation guarantee termination useful domains mohri describes algebraic shortest path problem operations form semiring domain edge weights close need every distributive lattice forms bounded semiring numeric domains used static analysis generally fail distributive use mohri framework directly guard body tail body aris update step step guard update step tail end fig find maximum value array example consider program figure figure shows program state aris executed scalar constraints left array constraints right branch simply add constraint scalar domain resulting add constraint scalar domain update resulting observe cases discover relationship indirectly via push scalar relations edge properties underlined invariants lost final result shown statement update instead replaced aris would first lift invariant singleton property fig array maximum example scalar domain push property segment allowing derive state figure shows program state replaced aris illustrates sufficient simply compute transitive closure must also lift properties scalar domain fully reduced state following properties must satisfied graph segments must internally consistent segment properties consistent scalar domain rri jss segment scalar domain must consistent scalar properties segment dui notice propagate constraints scalar component segments known tried propagate information segments would incorrectly derive soon segment determined empty one solution simply apply three rules fixed point reached guaranteed compute fully reduced state however direct construction conceptually clean suffers pragmatic issues relating termination efficiency shall see termination normalization process guaranteed terminate arbitrary lattices example assume analysis uses convex polyhedra consider state figure fixed point satisfy properties fig fixed point regions progressively reduced indefinitely let gray regions shown figure intention shares line segment assume start exploiting compute yielding polygon given topmost dashed line allows trim top portion compute trim region however changed reduce process asymptotically approaches greatest fixed point modify equations example slightly still valid approximation concrete state convergence immediate fixed point process clearly terminate interval domain possible interval drawn initial set also show guaranteed terminate octagons convex polyhedra proofs given appendix unfortunately yet general characterisation lattices termination guaranteed abstract transfer functions section describe abstract transfer functions necessary perform array content analysis language described section variable assignment effect scalar assignment abstract state follows behaviour underlying domain first project previous value assuming occur introduce new constraint scalar domain however project scalar domain must also update incoming outgoing edges becomes yrrx fss fss given otherwise assumption free always well founded however always transform program case frx xts fig array content graph analysis program given figure executing dashed edges indicate equality reality represent two edges labelled example consider program given figure immediately assignment state shown figure must first introduce new variable hold prior value transforming normalized graph statement shown figure handle statement must eliminate annotations edges corresponding resulting state note edge omitted diagram annotation state completes handling statement introducing new value scalar domain scalar domain discovers dashed edge normalizing state rule results becoming desired invariant projecting gives array reads array read aris relatively simple standard variable assignment must existentially quantify variable instead introducing relation scalar domain add constraint singleton segment transfer function may formulated yrrx arisss given rra xss otherwise normalization handles consequences scalar part array writes store value array index update corresponding edge property however sufficient singleton srraris fig state know elements initialized evaluate aris update edge however may also covered edge potentially overlapping edges must perform weak update taking join previous value new may covered edges previous analyses distinguish strong updates elements generally singleton segment updated given property weak updates elements segment may updated edge must updated possible current state possible edges feasible consider array state illustrated figure first elements initialized store aris update singleton property however segments may contain aris segments consistent index case must weakly updated annotations omitted identical annotation formulate yrraris fss given rra fss rra fss otherwise notice variable initially perform weak update segment however normalization procedure enforce consistency improving efficiency relaxation computing strongest matrix entries perform substantial amount redundant work include constraints scalar domain matrix entry constraints processed step computation ideal many irrelevant content properties abstract domain operations often proportional number constraints hence want construct relaxation discards irrelevant scalar properties shall use denote relaxation operation minimum must satisfy want make close possible keeping representation concise similar process constraint abduction however even relaxation may still lose relevant information example consider program state using domain octagons scalar property rrx yss segment properties rri ass rrk ass computing value gives expected rri ass however although relaxations rri ass rrk ass exact computing yields rri ass ass rri jss could avoid loss information conjoining scalar part step fixed point computation avoids loss information also defeats original goal reducing computation cost instead define conservative operation maintains enough additional information retain properties interest example consider analysis performed example modified operation gives rri ass rrk ass case compute get rri ass ass rri ass maintains property interest without repeatedly conjoining fixed point process another observation examples boundary constraints edge rri jss implied typically irrelevant segment properties unless rri nss rri nss rri nss fig solving set constraints strengthen derive rri nss use fail strengthen rri jss case edge must empty array content function index particularly troublesome domains explicitly store transitive closure constraints expend substantial computation maintaining consequences rri jss largely irrelevant lost join choose operator discards consequences edge boundaries must take particular care lose information third case mentioned consider state shown figure edge infeasible computing original fixed point obtain rri nss lifted scalar domain however rri nss obviously implied rri nss typically discarded property rri nss obtained never lifted scalar domain must therefore add following case normalization rules given section jss ensures scalar properties resulting infeasible segments included scalar domain since many elements may relaxation operator allows take advantage sparse matrix representations unfortunately aware existing general operations suitable computing must consider underlying lattice characteristics implementation complete section outlining suitable relaxation operators matrices dbms octagons relaxations polyhedra found appendix particularly difficult define analogous operators alternative domains value dbm octagon domain consists set constraints rrvi kss kss octagon construct relaxation computing transitive closure discarding constraints implied rri jss rri jss abstract states stored closed form simply collect constraints appearing rri jss avoid performing many implication tests instead collecting constraints involving variables experimental evaluation implemented analysis sparcolyzer prototype array content analyser language described section sparcolyzer implemented ocaml using fixpoint underlying domain implemented dbm domain customized operating sparse graphs experiments performed core duo ram running ubuntu linux set segment bounds using simple analysis collect variables may possibly indirectly involved computation array index tested sparcolyzer number array manipulation program fragments taken halbwachs added several additional fragments illustrate interesting properties analysis including members init rand family discussed section figure computation time instance given table instances taken include original reported runtimes although directly comparable experiments performed slower machine duo ghz ram using domain difference bound matrices disequalities ddbm implementation available tried reconstruct table compares runtimes two variants content domain approach halbwachs column naive variant uses direct implementation matrix represented array copy scalar domain stored matrix entry sparse variant stores row column set entries normalization operations need process entries definitely remain computes fixed point relaxed matrix rather directly sparse uses simple relaxation step discarding constraints involving array variable note sparse still iterates elements changes scalar domain occur change scalar domain may affect matrix element cases index variables highly expect partitionbased methods faster need compute closure transitive edges performance naive comparable instances partitions improves substantially complex instances sparse faster yet sometimes several orders magnitude still finds desired invariant two cases interesting compare behaviour sentinel first nonnull programs superficially appear quite similar cases set marker scan array find particular element however invariants necessary prove desired properties quite different http performance domains roughly equivalent absence explicit disequalities ddbm behaves identically dbm program init init offset init init init init arraymax copy partition hoare partition sentinel first nonnull naive sparse table analysis times seconds instances unable prove desired invariant marked aris ari aris else arjs ari arjs arjs aris ari aris arjs arjs arjs aris aris partition partition hoare fig quicksort partitioning done hoare case first nonnull require ares arss ares expressed using approaches gopan halbwachs store separate invariant total ordering amongst partition variables approach however handle disjunctive reasoning segment property quickly reaches consider partition variant quicksort partition step given shown figure imperative source language use allow loads inside conditionals reads hoisted outside corresponding loops example loop marked transformed shown point marked possible determine rrj iss rrarjs xss thus easy show rrarjs xss holds loop exit arjs hoisted version naive method prove invariant successfully property rrj iss rrej xss derived edge arjs exit loop rrj iss property gets extracted scalar domain get rrej xss rrej ass using sparse method however property discarded involves scalar variables invariant lost use original version without hoisting would unable prove invariant using either method express rrj iss rrarjs xss directly however method easily proves standard version partition hoare figure correct whether reads hoisted conclusion future work described new approach automatic discovery array properties inspired algebraic algorithms approach retains much expressiveness partitioning methods avoids need syntax dependence factorial partitioning step method successfully derive invariants range interesting array program fragments substantially faster approaches even modest numbers index variables several improvements could made performance analysis current implementation take advantage liveness information maintains entries content graph variables step clearly performance could improved eliminating variables matrix algorithms maintain shortest path information often improved storing transitive reduction graph domains distributive determine whether given edge must occur transitive reduction however would worth investigating whether maintaining transitive reduction would prove beneficial references blanchet cousot cousot feret mauborgne monniaux rival design implementation static program analyzer embedded software mogensen schmidt sudborough editors essence computation complexity analysis transformation volume lncs pages springer gupta sarkar abcd eliminating array bounds checks demand acm symposium programming language design implementation pldi pages acm press cousot cousot logozzo parametric segmentation functor fully automatic scalable array content analysis proceedings acm symposium principles programming languages pages acm press cousot halbwachs automatic discovery linear constraints among variables program proceedings fifth acm symposium principles programming languages pages acm press dill timing assumptions verification concurrent systems automatic verification methods finite state systems volume lncs pages springer dillig dillig aiken fluid updates beyond strong weak updates gordon editor proceedings european symposium programming volume lncs pages springer gopan reps sagiv framework numeric analysis array operations proceedings acm symposium principles programming languages pages acm press gulwani mccloskey tiwari lifting abstract interpreters quantified logical domains proceedings acm symposium principles programming languages pages acm press halbwachs discovering properties arrays simple programs acm symposium programming language design implementation pldi pages acm press maher herbrand constraint abduction proceedings ieee symposium logic computer science pages ieee comp octagon abstract domain symbolic computation mohri semiring frameworks algorithms problems journal automata languages combinatorics halbwachs abstract domain extending matrices disequality constraints cook podelski editors verification model checking abstract interpretation volume lncs pages springer schrijver theory linear integer programming wiley suzuki ishihata implementation array bound checker proceedings fourth acm symposium principles programming languages pages acm press pfenning eliminating array bound checking dependent types acm symposium programming language design implementation pldi pages acm press appendix termination proofs theorem computation terminates octagon domain proof consider initial set abstract states system inequalities form let bppxq denote bounding hyperplanes bpppx zqq bppx bppx time one evaluated bounding hyperplane equations bpp initial set bounding hyperplanes element finite iteration must tighten least one bounding plane tightening process must eventually terminate fact number descending steps bounded xpx case octagons projection introduce new bounding hyperplanes addition propagation rules affect termination theorem computation terminates convex polyhedron domain proof prove termination convex polyhedra similar fashion octagons consider initial set abstract states system inequalities form polyhedron decomposition theorem see polyhedron may generated finite set points set rays let denote set rays unit length direction variable vector component components given polyhedra xpx xpy results operations interest following properties none operations introduce rays xpx additional extreme points introduced application fixed point iteration exactly one three cases must occur abstract values remain set rays changes sets rays remain set extreme points changes case terminate set rays restricted iteration strictly descending case occur finitely many times assuming set rays remains fixed introduced extreme point must element xpx set finite step descending occur finitely many times without case occurring case must always occur bounded number steps occur finitely many times fixed point process must eventually terminate conclude process terminate commonly used relational numeric domains case polyhedra worth noting theorem provide bounds coefficients hyperplanes resulting polyhedra cases coefficients may grow quite large converging cause problems domains implemented machine arithmetic appendix relaxation polyhedra relaxation algorithm polyhedra follows intuition octagons wish collect transitive closure discard anything implied separately rri jss however polyhedra domain provides two difficulties computing transitive closure set linear constraints domain typically stored minimal set generators hyperplanes often bad idea general constraints bounded arity may exponentially constraints transitive closure original problem instead collect set constraints sufficient reconstruct constraints computed transitive closure would kept given constraints construct new constraint implied separately either resolution matched pair coefficients construct new constraint coefficient example consider constraints rrx rrz rrw may resolved contains term contains yields rrx however construct new constraints combining rather computing transitive closure explicitly given initial set interesting constraints constraints find constraints resolvable taking account direction previous continue process resolution steps add resolvable constraints found tcu rrk mss otherwise ptsignpkv rrk mss transpc given set linear constraints defining polyhedron construct initial set elements implied rri jss octagons avoid compute relaxation transpc performing implication tests instead initializing constraints containing variables example consider constraint interest rra additional constraints rry rrz initially trra yqu resolve rry since second step include adding point wqu return add anything either current value appendix partitions init randm large number partitions required init rand family necessarily obvious assuming must distinguish case array empty thus gives two base orderings nus tnus descriptions sets denote equivalence classes equivalence classes listed increasing order one possible value otherwise must distinguish possible relations resulting orderings follows nuti nus nus tnus tnus construct partitions introduce feasible locations partitions cases must explicitly distinguish element progression grows substantially faster yielding
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star altimeter waveform retracking using sparse representation conditional random fields ribana roscher bernd uebbing kusche sep institute geodesy geoinformation university bonn bonn germany abstract satellite radar altimetry one powerful techniques measuring sea surface height variations applications ranging operational oceanography climate research open oceans altimeter return waveforms generally correspond brown model inversion estimated shape parameters provide mean surface height wind speed however coastal areas inland waters waveform shape often distorted land influence resulting peaks fast decaying trailing edges result derived sea surface heights less accurate waveforms need reprocessed sophisticated algorithms end work suggests novel altimetry retracking star technique show star enables derivation sea surface heights open ocean well coastal regions least quality compared existing retracking methods larger number cycles thus retaining useful data novel elements method integrating information spatially temporally neighboring waveforms conditional random field approach detection relevant separated corrupted parts sparse representation approach identifying final best set sea surfaces heights multiple likely heights using dijkstra algorithm apply star data envisat missions study sites gulf trieste italy coastal region estuary bangladesh compare several established recent retracking methods well tide gauge data experiments suggest obtained sea surface heights significantly less affected outliers compared results obtained approaches keywords coastal oceans altimetry retracking sea surface heights conditional random fields sparse representation introduction several decades radar altimetry routinely used monitoring sea surface height ssh variations observed sshs play key role several applications ranging operational oceanography chelton tidal modeling wang savcenko bosch gravity estimation hwang serve important indicators climate research recently radar altimetry coastal zones gommenginger inland water bodies birkett beckley become topic increasing interest however applications one needs mitigate potentially significant land influence altimeter return signal altimeter instrument satellite emits spherically propagating radar pulse reflected surface range information inferred travel time cazenave addition returned signal energy measured time forming altimeter waveform shown ideal surface return waveform corresponds theoretical brown model brown estimated shape parameters model provide information mean ssh significant wave height swh amplitude strength reflected radar pulse used derive wind speed board satellite waveform signal sampled discrete epochs preprint submitted remote sensing environment september spacing travel time generally referred range gates chelton altimeter measurements refer individual point directly satellite rather footprint diameter several kilometers depending swh altitude altimetry mission illustrated fig return waveform open ocean consists three main parts first return energy radar pulse measured waveform contains thermal noise present radar systems soon front radar pulse hits wave crests altimeter footprint defined single point measured return energy begins rise afterwards pulse illuminates surface around initial point footprint becomes growing circle corresponds rapidly increasing signal energy measured altimeter waveform leading edge altimeter return waveform defined first energy return wave crests return energy radar pulse reached wave troughs fig point area footprint circle reaches maximum defined footprint plf chelton afterwards circle transforms annulus increasing inner outer radii fixed illuminated area corresponding signal energy measured outside plf referred trailing edge measured waveform fig slope trailing edge utilized derive information attitude altimeter satellite leading edge corresponds mean sea level wave crests wave troughs thus represents reference point range measurement algorithm board satellite tries position point inside range window fixed tracking range gate quartly range window consists fixed number range gates covering depending satellite mission positioned onboard tracker based prior information range however positioning always perfect located exactly tracking gate consequently requires reprocessing altimeter waveforms transmitted back earth procedure called retracking open ocean shape waveform close theoretical brown model slightly shifted tracking gate position easily corrected using ocean model retracker brown hayne deng ice surfaces waveform signal often contains two leading edges due radar signal partly penetrating upper snow layer specialized retracking algorithms developed martin however coastal areas waveform shape typically disturbed land influences altimeter footprint resulting peaks fast decaying trailing edges deviations coastal waveforms brown model lead conventional ocean retrackers generate diverging strongly biased estimates ssh peaks propagate along trailing edge towards leading edge altimeter ground track approaches coast lee order mitigate land influences waveform shape various tailored approaches proposed example methods seek model entire waveform halimi combined brown ocean model modeled asymmetric peak account land influences different approach dealing influence peaks retracked estimates first partition waveform step identify relevant parts waveform leading edge also possible peaks example hwang first identify relevant apply threshold retracking algorithm leads multiple equally likely ssh estimates location final estimate chosen based comparison height information way peaks appear outside relevant ignored recently retracking sshs inland water bodies uebbing combined approach hwang waveform model halimi suppress peaks trailing edge also account possible peaks close leading edge waveform could shown lead improved lake heights compared conventional methods different approach passaro suggested procedure similar previously published approach sandwell smith first step parameters amplitude range swh estimated second iteration fixed swh mean value derived first step amplitude range correction since swh estimations strongly correlated range correction leads improved sshs closer coast introduce novel method analysis sea surface heights altimetric waveforms utilize spatial information neighboring range gates within one waveform well temporal information neighboring waveforms along altimeter track altimetry retracker star applied altimetry data open ocean well coastal areas contributions twofold first analysis includes novel detection scheme knowledge first time integrates spatial well temporal information differs conventional detection algorithm hwang partition entire waveform separate instead identifying possible disjointed leading edges second order largely independent choice tuning hyper parameters within detection scheme derive multiple partitionings varying weight unary binary terms conditional random field leads range partitionings entire waveform subsequently point cloud equally likely sshs measurement position estimated using ocean model halimi employ dijkstra algorithm dijkstra find reasonably smooth sshs without resorting fitting detection scheme uses sparse representation approach return power range gates within one particular modeled weighted linear combination single common set basis waveforms derived synthetic brown waveforms concept applied many areas signal analysis wright study appears first uses radar altimetry sshs sea surface conditions wave height neither independent along tracks neighboring tracks sandwell smith spatial information used analysis example maus simultaneously processing sequence waveforms tracking travel times halimi smooth estimation altimetric parameters means integration spatial information carried different parts analysis latter two approaches example integrate spatial information neighboring waveform develop improved estimation algorithms retracking integrate information means conditional random field crf lafferty end introduce spatial relations return power range gates temporally neighboring waveforms neighboring waveforms within one pass cycle relations range gate power within single waveform way range gates relevant ssh estimation distinguished corrupted waveform parts since represented different linear combination basis see fig contrast halimi conditional random field used part algorithm enforce smooth estimation retracking parameters propose use conditional random field detection step subsequently retracking method applied identified individual deriving ssh thus effectively ignoring disturbing signals outside selected means approach could transferred future analysis altimetry paper organized follows first introduce altimetry data used validating method well two study sites sec describe detection integration knowledge employing crf sec moreover estimation single sshs determination best set sshs multiple heights using dijkstra algorithm explained sec sec compare performance detection existing method evaluate proposed analysis framework means analysing envisat waveforms coastal regions gulf trieste northern bay bengal coastal waters bangladesh sec concludes paper figure waveform disturbing peak caused land influences colored blue relevant part sea surface height determined detection illustrated green theoretical waveform model depicted orange data study sites data apply retracking method sensor geophysical data records sgdr mission well sgdrs european environmental satellite envisat missions ocean surface topography mission ostm launched mid succeeding mission orbit satellite flies near circular day repeat orbit altitude inclination separation groundtracks equal equator main instrument altimeter emits radar pulses aviso additional instruments microwave radiometer used derive wet troposphere correction well gps doris systems precise orbit determination rosmorduc sgdrs obtained archiving validation interpretation satellite oceanographic aviso team part centre national etudes spatiales cnes sgdrs sorted pass cycle including passes per cycle day repeat orbit utilize data passes beginning mission july cycle end cycle composition mission similar launched december successor mission launch june satellites flew orbit close distance allow intercalibration satellite missions months moved interleaved orbit located middle nominal orbit increase spatial resolution combined data january afterwards satellite moved drifting geodetic orbit passivated decommissioned july losing contact use data interleaved period february cycle january cycle sgdrs interleaved orbit acquired physical oceanography distributed active archive center ftp operated jet propulsion laboratory jpl part national aeronautics space administration nasa envisat satellite launched march succeeding mission orbit european space agency esa orbit repeat orbit altitude inclination allows satellite cover higher latitude regions compared higher spatial resolution separation equator cost longer repeat period days satellite carries total instruments doris positioning system microwave radiometer radar altimeter altimeter instrument importance rosmorduc altimeter dual frequency altimeter emitting radar pulses esa envisat sgdr data provided esa https utilize data june cycle september cycle extract envisat tracker range altitude waveforms needed retracking algorithm additionally atmospheric model corrections dry wet troposphere well ionosphere extracted sgdr data linearly interpolated high rate positions validation retracked coastal sshs tide gauge data hourly resolution university hawaii sea level center uhslc used hourly data uncorrected respect tidal inverse barometric effects apply corrections neither altimetry data tide gauge data remove influence corrections validation tide gauge data trieste italy available june december tide gauge station chittagong bangladesh data july til december additionally utilize openly available gdr datasets cycles envisat cycles mission include ales retracked ranges comparison distributed jpl validation significant wave height wind speed utilize model data erainterim reanalysis dee distributed european centre weather forecast ecmwf interpolated altimetry track study sites investigating quality proposed star algorithm selected two study sites located gulf triest coastal regions bangladesh sites include varying conditions including shallow coastal waters open ocean areas temporally submerged sand banks transition zones river estuaries ocean triest adriatic sea groundtracks pass well interleaved orbit pass chittagong bay bengal groundtracks nominal orbit pass well envisat pass figure location altimetry tracks tide gauges used validation retracked sshs triest first study site located northern adriatic sea part gulf venice includes gulf triest descending nominal orbit pass mission crosses study area fig crosses italian mainland ocean close city marano lagunare approx covers laguna marano short transition isola sant andrea track covers open ocean gulf venice transitioning croatian mainland last track croatian mainland runs close croatian coast distance less furthermore utilize data ascending pass interleaved orbit fig crosses study area first located open ocean groundtrack covers croatian mainland track covers open water reaches italian mainland track located less away slovenian mainland tide gauge station triest located harbor city triest gulf triest distance tide gauge station groundtrack closest point bangladesh second study site located northern bay bengal region right coast bangladesh region covered ascending nominal orbit pass crosses study area fig track coverage starts open ocean parts bay bengal reaches sandwip island track covers strip open water related estuary delta gbmd reaches bangladesh mainland latitudes sand banks located along track submerged high tide low tide additionally descending pass envisat mission runs almost parallel track crosses study site first reaches open water related gbmd track reaches sandwip island track transitions back open water afterwards covers remaining open ocean study area sandbanks mentioned might influence data acquisition low tide chittagong tide gauge station located chittagong harbor distance groundtrack closest point detection notation let consider consecutive waveforms collected along cycle contains return energy range gates arrange waveforms waveform represented set overlapping windowed waveforms windows centered comprising neighboring range gates illustrated fig single echo within framework detection means identifying deciding range gate indices define sparse representation models aim proposed approach optimally detect number models per waveform unknown needs determined detection process detection framework schematic detection framework illustrated fig input framework windowed waveforms synthetic brown waveforms brown waveforms basis waveforms collected dictionary used sparse modelling signals given input conditional random field crf formulated consists unary datadependent term computed sparse representation binary term enforcing smooth partitioning entire waveform variation weighting two terms yield various sets optimal indices resulting different partitionings entire waveform framework flexible regarding chosen methods sparse representation replaced methods correlation similarity measures following detailed explanations framework provided figure detection framework input given altimetric waveforms synthetic brown waveforms range gate represented windowed waveform range gate center point neighboring range gates illustrated blue violet orange areas conditional random field formulated consists unary term computed sparse representation binary term enforcing smooth partitioning waveform conditional random field graphical model constructed connecting temporally adjacent range gates well adjacent range gates within one waveform variation hyperparameters conditional random field result different partitionings entire waveform conditional random field order perform detection integrating information neighboring range gates make use conditional random field crf range gates represented graph range gate connected spatially adjacent range gates within one waveform temporally adjacent range gates along satellite ground track see fig basic idea assignment range gate represented windowed waveform model use sparse representation framework neighbored range gates assigned model summarized one therefore range gates one follow underlying model approach optimal partitioning minimizes energy functional unary term depending windowed waveforms describes agreement measured windowed waveform sparse representation model represented models identified activation indices indicating synthetic basis waveforms used signal reconstruction within specific model binary term depends activation indices well windowed waveforms indicating set direct neighbors range gate weight terms hyperparameter denoted approach assume set neighboring range gates modeled activation indices explained detail next paragraphs unary term sparse representation generally terms sparse coding olshausen field waveform represented linear combination basis waveforms collected dimensional dictionary reconstruction error solving problem means finding activation vector containing optimal coefficients whereas elements zero detection instead representing whole waveform signal windowed waveforms sparsely represented windowed basis waveforms case range gate represented windowed waveform assigned specific sparse linear combination windowed basis waveforms constituting best approximation detail windowed waveform sparsely represented using activations contains certain rows underlying dictionary see identical colored entities fig fig since dictionary comprises synthetic brown waveforms contain specific parts synthetic waveforms indices elements dictionary elements participating reconstruction activation indices used defining models cmp sec important note fixed consecutive waveforms used temporally neighboring windowed waveforms identical figure schematic illustration sparse representation windowed waveforms colors indicate different windowed waveforms independently sparsely represented colors dictionary indicating respective rows used reconstructing windowed waveform detection neighbored range gates represented dictionary elements grouped one optimal formulated argmin subject given activation indices set indices elements reconstruction error windowed waveform optimal activations vector collecting reconstruction errors possible sets dictionary elements optimization solved orthogonal matching pursuit omp tropp falls class greedy algorithms first dictionary element chosen one maximizes absolute value inner product dictionary element sample meant reconstructed maximize collinearity dictionary element chosen way however using current residual instead sample number used dictionary elements exceeds dictionary elements samples normalized alternative would exhaustive search combinations activation indices however would computationally challenging detail unary term energy functional penalizes reconstruction error given set activation indices difference activations sum describing agreement data specific sparse representation model abs first term normalized reconstruction error obtained activation indices unary second term normalized difference estimated activations terms normalized standard deviation values range gate given order ensure equal treatment range gates abs one penalization serves regularization constrain solution space reasonable results alternatively directly incorporated restrictive way introducing additional constraint often used remote sensing dictionary following properties purpose first elements high approximation ability second choice used dictionary elements reconstruction unique stable order build suitable dictionary set synthetically generated waveforms sampled relevant waveforms selected serve dictionary elements choose relevant synthetic waveforms selecting ones similar estimated waveforms dissimilar simulating dictionary elements parameters brown model brown waveform amplitude epoch sampled randomly probability density functions chosen resemble empirical distributions brown model parameters large set echoes envisat missions way proposed algorithm independent possible chosen altimetry mission binary term binary term serves incorporate prior knowledge spatial relations adjacent range gates within single waveform temporally consecutive waveforms within one cycle mentioned assume set neighboring range gates sparsely represented common set dictionary elements share activation indices set objective therefore prefer neighboring range gates similar characteristics reconstructed common set dictionary elements binary term given cos similarity measure cos relaxes constraint representation neighboring range gates order consider possible adaptions range window satellite tracker sparse conditional random field mentioned earlier evaluation possible sets activation indices computationally difficult however information needed crf find best set indices range gates overcome problem optimize crf greedy manner detail search optimal estimation activation indices iteration windowed waveform performed fixed crf application stopping single iteration identical usage correlation similarity unary term iteration optimal sets activation indices derived optimal set activation indices previous iteration final crf employed find optimal estimation activation indices iteration given minimizer energy abs entities correspond current iteration index omitted simplicity final optimal set activation indices given specific weight denoted sea surface height estimation study consider effect hyperparameter choice relative weight unary binary terms turned numerical experiments single choice provides optimal results different cycles different study sites altimetry missions therefore define range possible varying variation hyperparameter leads variation number detected therefore choose set reasonable hyperparameters words derive multiple partitionings ranging coarse partitioning includes fine one also captures small peaks separate ssh estimation use sufficiently large enough used retracking end several equally likely sshs measurement location could thought form point cloud finally dijkstra algorithm employed choose smoothest combination sshs finally single ssh location provided sea surface height context study define ssh hssh given hssh satellite altitude tracker range related fixed tracking gate provided altimeter data records atmospheric model corrections extracted sgdr data refer influence dry wet part troposphere well ionospheric influence signal retracking range correction derived retracking procedure described converting estimated epoch travel time range using speed light vacuum additional tidal corrections applied final heights selected reduce impact noise tidal corrections final height detection particular ocean tide correction introduce large noise component coastal areas corrupt selection final sshs dijkstra algorithm furthermore validate retracking results tide gauge data least hourly temporal resolution without tidal barometric corrections applied remove possible effects comparison purpose add relevant corrections hssh hssh sea state bias correction compute retracked swh additionally solid earth tide correction loading tide pole tide correction factor applies solid earth part pole tide correction ignoring part resulting ocean tides words apply ocean tide correction inverse barometric correction render hssh directly comparable high rate tide gauge data ocean model retracking retracking employ weighted ocean model halimi given erf exp travel time centered tracking gate thermal noise first range gates three fitted model parameters represent amplitude related backscatter rise time leading edge converted swh well epoch retracking gate refers position mid point leading edge additionally defined cos speed light vacuum radius earth deng pointing angle assumed zero application antenna beamwidth parameter defined brown computed beamwidth altimeter instrument algorithm finding best set sea surface heights problem discuss viewed optimization problem certain constraints given equally possible sshs need realize consistent ssh measurement position finding optimal ssh candidates applying algorithm remove outliers point cloud random sample consensus ransac algorithm fischler bolles assuming linear model ssh change direction sea level change linear thus apply ransac algorithm threshold moving window covering measured data selecting moving window cover make sure always include relatively large portion water especially transitions points deviate far linear model estimated ransac algorithm discarded resulting set accepted sshs used find shortest path estimating optimal sshs measurement location illustrated fig fig chose dijkstra algorithm dijkstra algorithms would also possible dijkstra method requires one choose edge weights individual connected nodes application chose height differences connected nodes edge weights thus favoring smaller height changes larger ones start end point dijkstra graph use first last sea surface height start end position figure schematic illustration optimal sea surface height estimation employing dijkstra algorithm dijkstra finds optimal path illustrated blue equally likely successively arranged sea surface heights edge weights derived difference heights nodes figure exemplary ssh pointcloud triest study site part proposed star algorithm black points selected ransac algorithm estimate linear model order discard outliers afterwards points within range linear model used dijkstra algorithm find best smoothest set sshs orange line results setting compare star algorithm existing retracking algorithms standard range derived retracking method provided sgdr data retracker martin equally weighted ocean model see furthermore specialized coastal algorithms adaptive leading edge subwaveform retracker ales passaro improved threshold retracker itr hwang latter combined threshold considered implementing star method described chose set neighborhood windowed waveforms number elements select values lead significantly increased computation times larger combination limited number dictionary elements might result less clearly defined unique combinations basis elements avoid finding optimal weighting parameter measurement region run five different choices result five partitionings total waveform ranging fine partitioning coarse one parameters computation basis elements generated based average estimated parameters application retracker current block waveforms waveforms per block framework mean epoch amplitude serve input basis element generation produce waveforms randomly varying mean parameters amplitude randomly varied mean value epoch randomly sampled using gaussian weighted distribution maximum mean value waveheight randomly sampled range common range waves coefficients obtained generated waveforms computed waveforms distinct kept form dictionary elements following utilize detected derive results star star algorithm itr implemented altimetry toolbox itr provides heights detected leading edge assumed first detected leading edge yield correct retracking parameters prior information heights sgdr sshs extracted gdr data ales retracked ranges extracted external gdr data sshs retracking methods processed way detection first compare detected star resulting five different weights method hwang arbitrarily chosen waveform measured coast croatia fig waveform parts detected part leading edge method presented hwang indicated white background contrast individual depicted alternating shades grey waveform five potential leading edges identified method hwang utilized derive potential sea surface heights comparing partitioning total waveform different weights star lower weights lead significantly increased number identified cases single contain one range gate since binary term weighted significantly lower unary term increased weight similarity constraint enforced size increases weight identified corresponds entire waveform leads including standard case retracking complete waveform derived point clouds next step compare resulting point clouds star orange points fig point cloud derived using hwang algorithm black points fig one exemplary cycle low tide conditions bangladesh site small bias open ocean identified methods related chosen threshold used itr sandbank near point clouds agree well however prior reaching sandbank points derived itr drop rapidly level approximately outside plot boundaries point cloud based star becomes less dense still preserves enough meaningful sshs beginning sandbank sandwip island bangladesh mainland star able derive sshs detected itr influenced land returns disturbing retrieved waveform results meaningful sshs detected low tide conditions case retracked sshs validation retracked sshs compare star sshs retracked sshs various conventional coastal methods top fig shows retracked sshs derived one arbitrarily selected cycle almost high tide trieste study site bottom part fig shows corresponding return waveforms measurement location open ocean find waveforms corresponding well theoretical brown model sshs retracking algorithms agree well however shows small bias compared retracking algorithms likely due nearest coast northern southern part study site sshs based sgdr indicate rapid drop sea level drop related influence peaks resulting land influence croatian italian mainlands coastal methods star ales figure partitioning using method weights set inserted waveform arbitrarily selected coastal waveform pass cycle grey shaded background depicts individual white background indicates part total waveform itr influenced peaks consider parts total waveform include peaks measurement positions southern coastal area sshs generally agree sshs star latter showing less noisy variations providing sshs right coast croatian mainland fig croatian coast sshs based ales itr show significant outliers related peak due land influence croatian mainland located close leading edge resulting biased estimations additionally smaller peaks preceding leading edge detected fig bottom identified potential leading edges itr algorithm thus result outliers due assumption utilize first identified leading edge itr algorithm similar behavior observed algorithm near peaks preceding leading edge lead significant outliers northern coastal area find good agreement sshs based ales itr star significant outliers ales last two kilometers nearest coast isola sant andrea drop ssh detected three methods itr showing strongest drop several meters sshs star drop less fig due relatively broad peaks located directly leading edge relatively shallow laguna marano methods except sgdr provide sea level similar open ocean level high tide despite waveforms consisting strong specular peak shapes fig bottom figure point cloud sshs resulting star compared itr hwang repeatability retracked sshs star method utilizes randomized dictionary decomposition therefore makes sense investigate whether leads uncertainty final sshs answer question conduct monte carlo study ran star algorithm times utilizing arbitrarily selected cycle pass mission resulting sshs shown fig top additional regions two coastal areas well open ocean bottom part figure displays corresponding root mean square difference rms ranging open ocean area also given additional range open ocean area variability along groundtrack range fig top repeatability star given rms less fig bottom southern coastal area variability range revealing slight increase sea level towards croatian coast fig top due land influence shape waveform visible parabolas fig bottom retracking results sensitive respect size individual detected nonetheless still find good repeatability region rms less fig bottom less compared along track variability significantly less compared relatively large outliers produced retracking algorithms region see fig top part variability current state algorithm due dijkstra algorithm employed find best sshs point cloud single points measurement location significant influence chosen path since allow edge connections neighboring locations example find standard deviation almost zero fig bottom top plot fig possible identify single point runs intersect since paths obtained dijkstra algorithm include point preceding succeeding sshs influenced tend close ssh point northern coastal region general variability range compared open ocean area repeatability shallow waters isola sant andrea similar croatian coast south rms less however find significant outliers runs related weak leading edge strong peaks close leading edge fig bottom laguna marano large variability detected fig due strong specular peak waveforms small changes detected significant impact figure top comparison ssh derived various retrackers along pass gulf trieste arbitrarily chosen cycle cycle distance nearest coastline dtc provided light gray bottom radargram corresponding waveforms inside study area waveform return power given color coded capped power visual reasons table median values different quality measures study site trieste see also top plots figs represents correlation percentage retained cycles rms retracker sgdr itr ales star open ocean croatian coast italian coast derived sshs comparison tide gauge data experiment compare ssh estimated using various retracking methods tide gauge data trieste italy chittagong bangladesh comparison utilize cycles data acquired described study sites see sec since period hourly tide gauge data well data obtained ales available tide gauge data july december interpolated times crossing cycle tide gauge data corrected tidal atmospheric pressure effects also apply corrections altimetry data employing remove outliers altimetric sshs deviation mean ssh evaluate correlation difference rms respect tide gauge data due large number available cycles set minimum number cycles required derive reliable correlations rms percentage cycles meet criteria shown fig median results following sections summarized tables figure repeatability star top sshs obtained runs cycle pass trieste study site additionally show zoomed coastal open ocean regions colors chosen randomly bottom corresponding root mean square difference rms derived runs open ocean region also provided zoomed table median values different quality measures study site coast bangladesh see also bottom plots figs represents correlation percentage retained cycles rms retracker sgdr itr ales star open ocean sandbank channel overall correlation investigate overall correlation tide gauge data position using available cycles fig results mission shown fig study site gulf trieste top coast bangladesh bottom open ocean area trieste study site fig top sshs obtained itr ales star show correlations tide gauge time series sgdr find correlations open ocean southward bangladesh study area fig bottom sshs derived considered retracking methods agree well south coastal shelf transitions towards deep ocean correlation begins drop border study site deep ocean correlation tide gauge data chittagong station drops rapidly already seen kusche central coastal shelf coast find correlations retracked sshs tide gauge data immediate coastal areas trieste bangladesh study sites algorithm used derive standard ranges sgdr data converge consequently sgdr sshs available regions towards croatian coast find decline level fig top agrees region shows rapid decline level coast also declines towards coast level italian coast find decline overall correlation retracked sshs tide gauge time series fig top isola sant andrea rapidly figure percentage total number available cycles applying outlier detection minimum number cycles requirement top study site gulf trieste bottom study site coast bangladesh distance nearest coastline dtc provided light gray decline level remains level increases level right coast agrees well laguna marano generally find lower correlation level even negative correlation southern part laguna retracked sshs starting coast sandwip island bangladesh study site overall correlations retrackers decline fig bottom sandbank area drops level shows moderate drop level indicates small drop achieved due automatically removing low tide conditions sandbank height fit conditions imposed ransac algorithm applying dijkstra algorithm see sec strip open water part estuary delta find nearly zero ales itr star found central part region lower correlations showing similar number retained cycles rms table related fact methods utilize total waveform mainly consists strong peak often lead divergence estimation heights algorithm influenced signal shape waveform leads varying parts waveform used derive amplitudes consequently inconsistencies deriving range correction number retained cycles compare number cycles ssh derived retracker utilized measurement location iteratively eliminating largest difference tide gauge time series time series correlation least achieved enables direct comparison individual retracking methods evaluation number retained cycles plotted position study areas fig open ocean regions study sites gulf trieste coast bangladesh figure correlation sshs derived several retracking algorithms including star method hourly tide gauge data top study site gulf trieste bottom study site coast bangladesh distance nearest coastline dtc provided light gray find good agreement nales nitr nstar retained cycles nsgdr find slightly lower level retainable cycles resulting divergence parameter estimation addition obvious time series altimetry derived sshs deep ocean parts bay bengal agree well time series obtained coastal shelf croatian coastal area retracking methods experience small downward peak near coast find rapid drop nales nitr level retained cycles nitr rising back retained cycles coast also detect drop followed rise coast sshs retained applying model coastal area nstar declines nearly open ocean coast italian coast find nearly retainable cycles approximately coast itr ales star nitr nales nstar steep drop less retained cycles detected near resulting land influences small islands separate laguna marano gulf trieste see also sec median percentage retainable cycles sgdr coastal area table quite low indicating difficulties reach convergence total waveform used retrackers provide larger number retainable cycles less affected land influences central part laguna marano generally find level retained cycles retrackers except star provides nstar bangladesh study site nsgdr start decline near towards coast sandwip island reach minimum level coast addition decrease nales nitr nstar detected star algorithm able retain least sshs available cycles region northern bay bengal tidal amplitude reach several meters leads significantly different land influences shape waveform low high tide consequently starts affect retrieved sshs coast nales nitr nstar starting decrease towards coast nales find strong drop figure percentage cycles retained achieve correlation least hourly tide gauge data total number available cycles top study site gulf trieste bottom study site coast bangladesh distance nearest coastline dtc provided light gray front sandbank region mentioned followed increase level nales coast nitr nstar drop level sandbank area itr star perform similar overall correlations suggest lower correlation itr region compared star due already removed sshs low tide cycles ransac algorithm part star strip open water sandwip island mainland bangladesh nales nitr nstar indicate three algorithms able retain available cycles central parts open water strip star able provide larger number cycles towards coastlines root mean square difference rms sshs derived individual retracking methods tide gauge data computed position provides additional quality measure fig apply criteria computation introduced sec open ocean generally find results similar analysis overall correlation agreeing well level trieste site bangladesh site slightly larger compared methods close croatian coast find increase significantly reaching large rms derived ales probably related small peaks front leading edge since ales selected first range gate end leading edge small peaks front leading edge lead difficulties parameter estimation star find smaller increase rms relative open ocean area northern part trieste site itr ales star keep level rms open ocean area distance nearest coast northern part bay bengal fig bottom sshs retracking methods become noisy northward minimum rms level figure root mean square difference rms derived locations applying selection criteria mentioned available cycles fig study sites top study site gulf trieste bottom study site coast bangladesh distance nearest coastline dtc provided light gray sandbank region rms obtained sgdr ales show rapid increase least rms based star exhibits relatively smaller increase level coast sandwip island level found small strip open water sandwip island mainland bangladesh similar level center region significant wave height besides range corrections use retracking model also allows retrieve swh swhs selected based heights chosen shortestpath algorithm theoretically would possible run dijkstra algorithm independently swh however think values consistent selected heights independently measured time series wave height wind speed available two study sites therefore compare temporal median rms number retained cycles good swh wind speed obtained individual retracking algorithms compare obtained results model data focus bangladesh study site since model data available gulf trieste area significant wave height fig shows median swh rms cycles bangladesh study site wave heights sgdr ales agree well model data open ocean star biased time median wave height values retrackers decline towards coast reaching minimum sandbank area contrary model data show decline open ocean rms values retrackers agree well ales star providing lowest figure swh compared model data top median cycles bottom rms cycles rms level model data suggests level coast rms values retrackers increase due sandbank influence low tide cycles low agreement retracked swh model data especially closer coast explained estimation retracking parameters utilized retrackers constrain swh parameter positive values since wave heights zero physically meaningful sometimes leads forcing wave height zero estimation individual waveforms since leading edge necessary swh parameter estimation represented limited amount range gates calm sea state conditions ales method problem appears less severe since two stage procedure swh estimated averaged first stage kept fixed second stage retrackers star problem occur often case selected contains leading edge relatively small case zero wave heights utilized derive median rms value sgdr ales star agree well model data fig table sandbank area channel sandwip island main land sgdr quality decreases due low number available cycles fig bottom zero wave heights occur cycles calm sea state conditions bangladesh study site especially coastal areas similar observations made trieste study site future star may extended procedure similar ales order counter effects also applied method reduced sar rdsar waveforms conventional altimetry like waveforms derived sar signal rdsar data data showed problem occur applied derivation reduced sar waveforms due doubling number range gates available allows better estimation swh shown wind speed wind speed derived estimated amplitudes utilizing model gourrion described aviso model input consists retracked swh utilized swh order keep wind speed estimate unbiased problems swh estimation similar swh display median rms retracker erainterim model data measurement position open ocean median wind speed rms sgdr itr star agree well wind speed derived model data fig table however sgdr figure swh compared model data top median cycles bottom rms cycles removed table median values swh median rms different regions along track bangladesh study site removed denotes median rms retracker sgdr ales star open ocean sandbank channel slightly biased model data itr star slightly biased model data ales shows stronger bias respect model data towards coast waveforms start get influenced land returns leading peaks moving along trailing edge towards leading edge retrackers utilize full waveform sgdr return amplitude estimates biased peak influence thus longer provide reliable wind speed method itr ales star also show slight decline wind speed sandbank area one finds mostly specular peak waveforms derived wind speed especially low tide longer meaningful center channel sandwip island bangladesh main land ales star able provide meaningful wind speed reason bias wind speed sgdr star due chosen estimation model sgdr data estimation method employed utilizing parameter model amarouche fourth parameter angle derived slope trailing edge assumed zero models angle influences estimation amplitude amarouche lead general bias amplitude case angle different zero derived angle slightly positive leads smaller amplitude consequently higher estimated wind speed even small changes significant influence derived wind speed however methods star feasible try estimate angle since include enough range gates trailing edge reliable estimation figure wind speed compared model data top median cycles bottom rms cycles table median values wind speed median rms different regions along track bangladesh study site denotes median rms retracker sgdr itr ales star open ocean sandbank channel application envisat data investigate application star altimetry data interleaved period trieste site fig envisat data banlgadesh site fig however interleaved available comparison focus number cycles retained reach correlation fig top well rms position fig bottom found cycles overlapped available time period tide gauge data trieste station found cycles envisat data overlapped tide gauge period available chittagong station utilizing data mission open ocean trieste study site itr star show retained cycles well significantly smaller rms compared sshs coastal area last front croatian coast star derived sshs fit well tide gauge data fig crossing parts croatian mainland track transitions back ocean find itr star retaining significantly cycles compared general level retained cycles lower due signal losses happened coasts croatia italy nine interleaved cycles italian coast itr star able maintain high quality sshs coastal area decreases slightly last coast additionally utilized envisat track experiments crosses trieste study site however overlapping period available tide gauge figure top percentage cycles retained achieve correlation least hourly tide gauge data total number available cycles interleaved mission bottom root mean square difference rms derived locations distance nearest coastline dtc provided light gray data less one year deriving correlations rms short period containing cycles envisat repeat orbit would meaningful nonetheless sshs derived cycles envisat data compared well available tide gauge data shown study site coast bangladesh found overlap period years cycles envisat data chittagong tide gauge allows derive meaningful correlations rms differences open ocean area number retained cycles rms agree well considered retracking methods bases sshs slightly noisy fig envisat track reaches sandwip island differences individual retracking methods become evident ratios retained cycles sgdr show rapid drop less sandbank area gradual increase towards coast sandwip island level respectively behavior combined strong increase rms differences increase similar behavior observed itr show smaller drop number retained cycles last towards coast level corresponding rms increases significantly values greater ales data available last front sandwip island star shows smallest drop number retained cycles recovers back level close sandwip island small strip open water sandwip island mainland bangladesh coastal retrackers itr ales star retain available cycles providing sshs fit well tide gauge data sgdr able provide retained cycles exhibit significantly higher rmss region conclusion novel method analyzing altimetric waveforms deriving sea surface heights swh sigmanought suggested proposed technique partitions total waveform individual figure top percentage cycles retained achieve correlation least hourly tide gauge data total number available cycles envisat mission bottom root mean square differences rms derived locations distance nearest coastline dtc provided light gray waveforms analyzed combination existing retracking models sea surface heights provided star found least quality better compared existing conventional coastal retracking methods open ocean well coastal regions addition correlations tide gauge data revealed generally usable cycles close coast combination lower root mean square differences compared existing methods course depending retracking model combined derived possible derive significant wave height backscatter way comparison derived hwang method reveal good correspondence identified parts waveform found influence random component star ssh results level less open ocean coastal regions range modifications applied conventional retracking algorithms including biases different retracking methods different weighting schemes varying estimators effects considered order centimeters sea surface heights derived star algorithm extensively validated data compared five independent available retracking methods well hourly tide gauge measurements study sites gulf trieste italy coast bangladesh found varying surface conditions including deep open ocean shallow coastal waters temporally submerged sandbanks transition zones river estuaries ocean consequently deriving five partitionings total waveform enabled star algorithm handle larger variety waveform shapes compared existing coastal retracking algorithms examination estimated swh wind speed revealed good agreement retracking algorithms well model data furthermore applied star altimetry data interleaved period well envisat also resulted significant improvements quality coastal sea surface heights confident star method enable wide range studies including comprehensive validation significant wave height context also consider improved selection final retacking results considering swh etc instead utilizing sshs addition algorithm improved tuning hyperparameters control resolution well extending dijkstra algorithm reduce impact potentially sea surface heights neighboring measurement locations derived combined different available waveform models order adapt extend concept regions rivers lakes one might also consider slightly different approach using partitioning derive weighting schemes retracking whole waveform acknowledgements acknowledge funding project also thank associate editor three anonymous reviewers constructive comments helped significantly improve manuscript references amarouche thibaut zazanife dumont vincent steunou improving ground retracking better account attitude effects marine geodesy aviso ostm products handbook cnes nasa http issue plaza dobigeon parente gader chanussot hyperspectral unmixing overview geometrical statistical sparse approaches ieee journal selected topics applied earth observations remote sensing birkett beckley investigating performance radar altimeter lakes reservoirs marine geodesy brown average impulse response rough surface applications ieee trans antennas chelton ries haines callahan satellite altimetry cazenave eds satellite altimetry earth sciences handbook techniques applications chapter academic press chelton walsh macarthur pulse compression sea level tracking satellite altimetry atmos oceanic dee uppala simmons berrisford poli kobayashi andrae balmaseda balsamo bauer bechtold beljaars van berg bidlot bormann delsol dragani fuentes geer haimberger healy hersbach isaksen matricardi mcnally morcrette park peubey rosnay tavolato vitart reanalysis configuration performance data assimilation system quarterly journal royal meteorological society doi http deng improvement geodetic parameter estimation coastal regions satellite radar altimetry thesis curtin university technology dijkstra note two problems connexion graphs numerische mathematik esa envisat product handbook european space agency esa https url https issue dinardo scharroo becker benveniste weiss german bight validation altimeter data sar mode advances space research fischler bolles random sample consensus paradigm model fitting applications image analysis automated cartography communications acm cazenave eds satellite altimetry earth sciences handbook techniques applications academic press gommenginger thibaut quartly deng challenor gao retracking altimeter waveforms near coasts review retracking methods applications coastal waveforms vignudelli kostianoy cipollini benveniste eds coastal altimetry berlin heidelberg gourrion vandermark bailey chapron gommenginger challenor srokosz wind speed algorithm altimeters atmos oceanic halimi mailhes tourneret snoussi bayesian estimation smooth altimetric parameters application conventional altimetry ieee transactions geoscience remote sensing halimi mailhes tourneret thibaut boy parameter estimation peaky altimetric waveforms ieee trans geosci remote sensing hayne radar altimeter mean return waveforms ocean surface scattering ieee trans antennas hwang guo deng hsu liu coastal gravity anomalies retracked altimetry improvement limitation role airborne gravity data journal geodesy hwang kao parsons global derivation marine gravity anomalies seasat geosat altimeter data international geophysical journal kusche uebbing rietbroek shum khan sea level budget bay bengal grace altimetry journal geophysical research oceans lafferty mccallum pereira conditional random fields probabilistic models segmenting labeling sequence data proc international conference machine learning url http lee shum emery calmant deng kuo roesler validation altimeter data waveform retracking california coastal ocean marine geodesy martin zwally brenner bindschadler analysis retracking continental ice sheet radar altimeter waveforms journal geophysical research oceans maus green fairhead improved resolution retracked satellite altimeter waveforms geophysical journal international olshausen field sparse coding overcomplete basis set strategy employed vision research url http doi http passaro cipollini vignudelli quartly snaith ales adaptive subwaveform retracker coastal open ocean altimetry remote sensing environment quartly srokosz mcmillan analyzing altimeter artifacts statistical properties ocean waveforms journal atmospheric oceanic technology rosmorduc benveniste bronner dinardo lauret maheu milagro picot radar altimetry tutorial http url http benveniste esa picot cnes sandwell smith marine gravity anomaly geosat satellite altimetry journal geophysical research sandwell smith retracking altimeter waveforms optimal gravity field recovery physical journal international savcenko bosch empirical ocean tide model satellite altimetry dgfi report tropp gilbert strauss algorithms simultaneous sparse approximation part greedy pursuit signal processing uebbing kusche forootan waveform retracking improving level estimations altimetry observations african lakes ieee transactions geoscience remote sensing wang ocean tide modeling southern ocean technical report department civil environmental engineering geodetic science ohio state university usa columbus ohio wright mairal sapiro huang yan sparse representation computer vision pattern recognition proceedings ieee
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dec simple introduction karmarkar algorithm linear programming sanjeev dept computer science engineering indian institute technology kanpur december abstract extremely simple description karmarkar algorithm technical terms given introduction simple description karmarkar algorithm together analysis given paper knowledge simple algebra vector dot product matrices assumed even though method described several books analysis either left fairly complicated show presentation roos terlaky vial simplified mainly using essential notation allmost results paper standard textbooks see references let matrix rank vector ones karmarkar problem min subject constraints assumed optimal value zero ones vector feasible either find point cost show none exist first scale variables problem becomes min equivalently min subject drop primes problem ssax min subject constraints remark problem trivial hence assume assume equality constraints linearly independent else eliminate redundant rows need definitions standard simplex consists points dimensions centre simplex let radius outer sphere smallest sphere containing standard simplex circumscribes standard simplex distance one corner point see figure example three dimensions standard simplex say thus let radius another largest sphere centred completely inside standard simplex inscribed inside standard simplex sphere tangent face face one coordinate symmetry coordinates point contact say point contact standard simplex hence take starting point assumption feasible minimise objective function smaller sphere call inner sphere centre radius less see section chosen let assume minimum occurs point next iteration take starting point problem minimisation sphere discussed next section section finally point mapped process repeated details mapping section section shown objective value next point fraction initial point mathematical preliminaries let point possible hyper plane unit vector normal point plane vector dot product zero writing full equivalently thus normal direction vector equation sphere centre let point plane know want perpendicular plane constant consider problem min subject two points sphere see figure example two dimensions planes parallel given plane one corresponding maxima minima points say direction unit vector direction length thus minimise sphere start centre take step length direction less informally points sphere optimal solution obtained taking step length radius sphere centre direction seen considering parallel planes point minimum attained point contact sphere tangent plane next consider problem min subject points common intersection sphere plane optimal let orthogonal projection onto plane linear combination rows objective function intersection sphere plane constant feasible points also optimal solution obtained taking step length equal radius lowerdimensional sphere intersection sphere plane centre direction solution inner sphere let consider problem min subject minimising points inner sphere inner sphere completely inside standard simplex drop constraint also problem becomes min point feasible lies plane moreover centre inner sphere lies intersection inner sphere sphere radius lower dimension intersection sphere plane containing centre circle radius centre minimum value linear function sphere point lower one linear function touches sphere radius vector point perpendicular plane let assume unit vector direction point minimum attained radius thus outer sphere say inner sphere see determine later see section outer sphere completely contains solution space minimum value objective function smaller equal actual optimal value zero thus assuming minimum value thus cre inner sphere inside simplex value objective function less simplex optimal value simplex zero optimal value inner sphere minimum value occurs inner sphere value objective function get thus start initial solution next solution improvement value objective function factor initial solution karmarkar transform algorithm like map new point repeat process thus define transform least component transform less sum coordinates thus range standard simplex moreover inverse transformation thus transformation standard simplex remark also seen directly say moreover number nai thus modified problem assume applying transform point mapped let diagonal matrix diagonal entries diag feasible point positive satisfying saw unique point depends remark implies thus xzii equation equivalently aij aij equivalently aij objective function optimal value zero follows optimal value transformed objective function also replacing using diag transformed problem min subject moreover feasible point equivalently aze thus feasible thus repeat previous method instead instead words minimise modified objective function inscribed sphere radius modified problem min subject azx result projection matrix assume matrix rank said iff rows linearly independent row vector size one solution thus matrix column vector size implies use fact matrix aat rank let matrix hence represents set equations equation hyperplane dimensions let vector size like project onto lower dimension intersection hyperplane projection wish write best possible rows projection vector size written matrix matrix product aat square matrix let matrix size consider equation aat get yaat dot product real vectors matrices means term individually zero implies identically zero thus matrix aat rank invertible error projection rejection also vector size error hyperplane perpendicular hyperplanes thus want thus aat aat aat hence aat aat thus summarise projection matrix aat rejection matrix get part perpendicular hyperplanes aat algorithm start algorithm find optimal value inscribed sphere radius see details step later typical iteration next apply transformation map need transformation gets mapped modify find new say optimal value inscribed sphere radius remark know transformed value transform actually applied values updated apply inverse transformation map original space get value next iteration formal algorithm value fixed section initialise main step return current solution desired accuracy let diag let ndy function analysis define potential function log log arithmetic mean greater equal geometric mean taking logs log log log exp next observe log log log log log log log log let positive vector standard simplex let nxi moreover log log finally log log log log log log log log log log log cct ratio original transformed problems saw reduces least thus log log show constant continuing analysis need results algebra function section logs base assume log let log log define log observe hence claim greater log defined proof know without loss generality assume coefficients sides equal thus need compare thus comparing equivalently expressions terms inside brackets positive right hand side negative claim greater log defined proof induction let claim induction hypothesis proof follows adding two inequalities analysis continued log hence log log log log log log log unit vector feasible hence log last inequality follows corollary defined choose log log log log log hence potential decreases fixed amount iteration iterations log get log inside standard simplex exp log exp differentiating equating get thus maximum value stop soon error get log exp log log log log log thus log iterations algorithm finds feasible point acknowledgments many thanks students batch valuable feedback questions comments earlier version references mokhtar bazaraa john jarvis hanif sherali linear programming network flows edition wiley donald goldfarb michael linear programming chapter optimization edited nemhauser rinooy kan todd handbooks operations research management science volume pages elsevier linear programming birkhauser katta murty chapter linear complementarity linear nonlinear programming web book http narendra karmarkar new polynomial time algorithm linear programming combinatorica vol terlaky vial interior point methods linear optimization springer linear programming modern integrated analysis kluwer hamdy taha operations research introduction edition pearson lyu lecture notes linear algebra lecture chap projection projection matrix institute space science national central university taiwan spring http
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nov word order problems automata groups laurent bartholdi ivan mitrofanov abstract prove word problem undecidable functionally recursive groups order problem undecidable automata groups even assumption contracting introduction let finite set consider group acting faithfully set words means every acts form depending encode map furthermore finitely generated say finite set quotient free group action may described finite data namely lift restriction generators finitely generated group given manner called functionally recursive call group presented write call asynchronous transducer large classes finitely generated groups presented functionally recursive ones notably iterated monodromy groups nekrashevych automata groups mentioned even though map completely determines action therefore unclear much known first result negative theorem algorithm given determines whether automata groups assume functionally recursive group action elements length generating set replacing map takes form call finite state transducer group called automata group form notorious class groups containing finitely generated linear groups well infinite torsion groups grigorchuk group groups grigorchuk group also group intermediate used settle milnor problem group growth action may conveniently described finite labeled graph called moore diagram consider directed graph vertex set date october work supported raction grant laurent bartholdi ivan mitrofanov figure transducer generating grigorchuk group edge labeled whenever action determined follows given find unique path starting whose first label letters read let second label letters see figure graph describing grigorchuk group every element say represented word length admits similar description using graph vertex set word represents identity every vertex reachable outgoing edges labels tpa follows word problem decidable even belongs linspace therefore exptime much known consider order problem determine order element raised end last century sidki grigorchuk nekrashevych sushchansky problem gillibert announced solution july proof appears theorem algorithm given determines order namely cardinality xsy worse action uncomputable following sense consider natural extension action following variants theorems theorem algorithm given determines whether fixed theorem algorithm given determines cardinality orbit xsy finally results theorems made uniform follows theorem functionally recursive group recursive theorem automata group two states set stn finite orderu recursive word order problems automata groups contracting groups assume functionally recursive group action elements shorter generating set sense constants replacing set words length also thus defined subclass automata groups called contracting automata groups see precise definition word problem decidable logspace therefore polytime see however order orbit order problems remain unsolvable restricted class theorem theorem transducers constructed theorems may assumed generate contracting groups sketch proofs encode minsky machines functionally recursive groups minsky machines see restricted turing machines two tapes may move tapes sense tape end may write equivalently finite state automata equipped two counters values may incremented decremented tested cases encode machine state counter values word functionally recursive group containing elements element state machine action group devised machine evolves state recursive action given image prescribed ray records computational steps machine started particular whether machine reached final state construct auxiliary element acts sequences containing trace final state fixes original ray machine never reaches final state taking commutator last element acting neighbourhood original ray yields expression trivial machine never reaches final state inherently sometimes output transducer longer input machine increments first counter transducer must replace obtain automata group transducer consume power input word incrementation counter may performed erasing every second every second block element may arranged finite order machine reaches final state tilings results functionally recursive groups transducers may also interpreted terms tilings let finite set colours set wang tiles valid tiling map tpx yqn tpx tpx yqe tpx yqw berger showed undecidable determine given whether exists valid tiling improved call set tiles every exists one tile exists precisely one tile conditions lukkarila showed undecidability result holds even restriction clearly tileset tiles uniquely first quadrant choice colours axes result order problem following translation tilings consider tilings upper tpx following problem undecidable even tilesets given integer laurent bartholdi ivan mitrofanov every tiling upper horizontal axis horizontally indeed given set whenever build tile respectively also build tiles zps tiling problem satisfied finite order word problem may also translated tiling problem hyperbolic space tileset lattice tiling map xqqe tiles visualized pentagons assembling tiling hyperbolic plane invariant transformations following problem undecidable even tilesets given every tiling edge identical labels boundary indeed subdividing inserting empty state may assume map describing functionally recursive group satisfies tiles defined history links established since beginning undecidable problems theoretical computer science halting turing machines algebra decidability properties algebraic objects minsky machines simplicity early recognized useful tools correspondence see gurevich work identities semigroups automata semigroups defined quite similarly automata groups one merely drops requirement action invertible maps decision problems extensively studied within class automata semigroups gillibert proved order problem unsolvable class proof based undecidability wang tiling problem harnesses kari solution nilpotency problem cellular automata usually serious difficulties converting solution semigroups one groups particular tilings heart gillibert construction give fundamentally transformations hand direct approach order problem succeeded restricted class bounded automata groups bondarenko sidki zapata prove solvable order problem gillibert announced undecidability order problem automata groups work uses simulation arbitrary turing machines transducers via cellular automata functionally recursive groups minsky machines theorems proven embedding minsky machine computations functionally recursive groups let recall precisely definition machines definition minsky machine computational device equipped two integer counters finite amount additional memory finite word order problems automata groups set states initial state final state state instruction following kind iii vii viii valid valid use style operator meaning else turned state counters initialize psi determined psi using prescribed rules moment stops otherwise runs forever recall main result mealy machines namely powerful turing machines proposition algorithm given minksy machine determine whether stops universal minsky machine stops turned state nqu recursive also note one instructions iii necessary presence one one one vii viii one necessary minimal sets instructions iii vii viii vii iii proof theorem let minsky machine stateset without loss generality assume instructions type construct functionally recursive group presented sets given follows generating set consists elements state type element state type three elements alphabet consists four letters state state state type letter type two letters type five letters map given denoting empty word whenever value unspecified take mean laurent bartholdi ivan mitrofanov states put sztuu put put instruction psi psj type put psj instruction psi psj type written position following table element pai pai input letter pxbi pbi instruction psi type put psk theorem follows undecidability halting problem minsky machines proposition following proposition consider infinite sequence minsky machine halt action satisfies xyq proof encode states elements word psi corresponds state psi convenient write form manner computation functionally recursive action given sequence exchanges letters words check following equalities psi psj instruction type psi qtpsi psj qtpsj indeed claim follows reverse psi psj psi psj instruction type psi qtpsi psj qtpsj indeed first check xpxbi word order problems automata groups obtained word two blocks blocks time letter multiplied left word size first block halve size second one double xbi xbi recalling claim proven maxpm instruction type qtpsk psi qtpsi psj qtpsj psi qtpsi psk indeed first case xqy second case recalling claim proven follows halt xyq conversely halts exist qtps tps tps case xyq computations best carried dual moore diagram see figure directed labeled graph vertex set edge labeled whenever one checks equality finding path starting input label endpoint path output label proof theorem yet used letter state theorem follows following proposition minsky machine halts proof element acts follows scans longest prefix exchanges prefix write proposition know fixes halt assume first halt fact also fixes supports disjoint laurent bartholdi ivan mitrofanov pbi pai psi type type pbi xbi psi type figure dual moore diagram used proof theorem assume next halt without loss generality may assume stop immediately since commutator acts particular proof theorem consider universal minsky machine namely one emulates arbitrary turing machines encoded integer started state set halts started state recursive see proposition therefore theorem follows considering group elements set words recursive subset equal recursive automata groups minsky machines proof theorem let minsky machine stateset without loss generality assume instructions type iii vii viii defined beginning section siii siv svii sviii consider transducer stateset alphabet tiiii ivi viij viiij word order problems automata groups state structure transducer given map first described table position state identity instructions type iii zsiii input letter state state instructions type zsiv input letter pps applies instruction type roles switched instruction type vii zsvii input letter applies instruction type viii roles switched note treated state tables theorem follows undecidability halting problem minsky machines following proposition minsky machine constructed halts element finite order proof set denote cpgq symmetric conjugacy class cpgq given symmetric conjugacy class choose representative let decomposition cycles action choose representatives pai phi easy see collection symmetric conjugacy class tcphi independent choice construct directed whose vertices symmetric conjugacy classes conjugacy class edges starting ending respectively labels graph essentially appears solution order problem bounded automata laurent bartholdi ivan mitrofanov lemma order least common multiple along paths starting cpgq product labels along path proof consider path starting cpgq labels going vertices cpgs orbit length order multiple furthermore fixes orbit acts sequences start orbit recursively order multiple order multiple particular paths arbitrarily large product labels infinite order conversely infinite order arbitrarily long orbits paths arbitrarily large product labels least common multiple path labels edges paths starting cpg labeled fixes every sequence therefore let compute subgraph spanned xyq computations helpful picture operation transducer means dual moore diagram see figure given compute primitive cycles whose input label power read corresponding output label map symmetric conjugacy classes cpgq tcphi first note direct inspection commute follows tracing path graphs noting always induce trivial permutation output either trivial conjugate claim transition machine conjugacy class least one arrow possibly arrows also claim type arrows labels arrows cps arrows see machine halts every path starting xyq finite number labels shows order finite hand machine halt path infinitely many labels minsky machine decrease counters infinitely many times row infinite order note transducer property pli pli tiii vii viiiu also note pli pli whenever instruction type using prove type tgn fixes orbit tli output indeed tgn pgn tgn qgn use induction follows ptxm fixes outputs arrow cptxm label let first restrict orbit tiiii consider sxm instruction type iii acts product two cycles output label first cycle starting output starting second cycle therefore two arrows required consider txm instruction different type considered arrows labels word order problems automata groups figure dual moore diagram used proof theorem restrict next orbit tivi consider case need consider cases txm first suppose perform instruction type second already considered element fixes outputs respectively hence arrows labels restrict next orbit tviii perform computations result following table gpg cycles output starting first element cycle laurent bartholdi ivan mitrofanov therefore two arrows labels arrow label two arrows label arrow label orbits tvi tviiii investigated way tivi tviii respectively proof theorem consider universal minsky machine namely one emulates arbitrary turing machines encoded integer started state set halts started state recursive see proposition therefore theorem follows considering group elements proof theorem let minsky machine stateset without loss generality assume instructions type iii associate transducer stateset alphabet iiii ixj state structure transducer given map state identity instructions psi type iii input letter state every instruction another type acts instructions psj type input letter every instruction another type acts applies every instruction psk type roles switched claim orbit finite machine stops set construct directed graph whose vertices elements consider action minimal fixes graph put edge label size orbit finite number equal product labels along path starting claim instruction edge label edge checked dual moore diagram see figure word order problems automata groups psk psi psk psj psk psj psj psj psk psi figure dual moore diagram used proof theorem first note commute instruction type iii orbit action edge labeled consider next instruction type two cases orbit output orbit output means edge labeled edge labeled naturally applies instructions type finally element fixes edge labeled contracting automata proof theorem finally explain make transducers previous subsections contracting expand definition introduction definition definition let group state unique element puvqg namely action tails sequences starting group contracting exists finite subset exists npgq whenever npgq minimal subset satisfying definition called nucleus particular one induces automaton still written replacing larger npgq thus making transducer process letters time laurent bartholdi ivan mitrofanov one may also assume transducer extra property called nuclear note probably undecidable whether group contracting easy decide whether transducer nuclear minimizing composite transducer find set words equal nuclear exists precise form theorem theorem algorithm given nuclear transducer determines cardinality orbit xsy algorithm given nuclear transducer determines order note group changed operations replacing nuclear contracting sense introduction since word metric defined conversely one may take see contracting sense definition lemma let transducer constant every reduced path length dual moore diagram contains letter along output contracting proof consider represent word length factor every computed following dual moore diagram path starting label input output label along path hypothesis time segment read letter produced shall modify transducers composing appropriate machines recall general definition let transducers stateset composition transducer alphabet given given transducer stateset alphabet write freely identify words value require commute every consider transducer alphabet transitions note drawing dual moore diagram deleting transitions output paths input output length form wsi word involving note also word form conjugate word order problems automata groups consider also transducer alphabet transitions ppa cqq ppa cqq ppa cqq note dual moore diagram paths input label form shorter output label soon note also word form ppa cqq permutation particular conjugate furthermore get proposition hypotheses transducer generates contracting group whenever psi original transducer relation ppj psi aqq either equal conjugate furthermore proof applying transducers words get shortened form wpx yqsj get shortened soon using fact commute follows contracting consider transitions psi transducer input letter produces psi input letter position produces conjugate psi input letters produces feed psi transducer input letter produces input letters produces conjugate feed finally conclude proof ready finish proof theorem constructed integerlabeled graph transducer whose vertices elements theorem symmetric conjugacy classes theorem proposition transducer contracting let check order problems equivalent graph set vertices graph proposition shows new graph set outgoing edges element form labels multiplied possibly new edges since old graph loops elements infinite order infinite order orbit infinite action orbit infinite action may finally replacing stateset assume nuclear outlook proved article undecidability order problem automata groups namely groups transformations generated transducer transducer belongs restricted class may well order problem becomes decidable particular importance laurent bartholdi ivan mitrofanov transducers polynomial growth transducer represented graph figure let denote number paths length end state bounded function case grigorchuk group order problem solvable see happens bounded linear function polynomial degree groups generated transducers considered sidki reset transducers transducers sqq functions namely state reached transducer independent original state transducers intimately connected tilings kari construction gillibert proved order problem unsolvable semigroups reset automata solvable groups reset automata reversible transducers transducers whose dual invertible related reversible turing minsky machines order problem solvable groups generated reversible automata bireversible transducers transducers transducers obtained inverting permuting stateset alphabet remain transducers give another point view square complexes tiling plane squares whose labels expect undecidable whether functionally recursive group actually automata group larger generating set whether automata group contracting even whether contracting group finite related questions semigroups known undecidable constructions similar references ali akhavi ines klimann sylvain lombardy jean mairesse matthieu picantin finiteness problem automaton semi groups internat algebra comput doi robert berger undecidability domino problem mem amer math soc ievgen bondarenko natalia bondarenko said sidki flavia zapata conjugacy problem automorphisms regular rooted trees groups geom dyn doi appendix jungers andrew brunner said sidki automorphism group binary tree algebra pierre gillibert finiteness problem automaton semigroups undecidable internat algebra comput doi automaton group undecidable order engel problems available rostislav grigorchuk burnside problem periodic groups funkcional anal english translation functional anal appl milnor problem group growth dokl akad nauk sssr rostislav grigorchuk volodymyr nekrashevych automata dynamical sysytems groups prosi narain gupta said sidki burnside problem periodic groups math problem equality words certain classes semigroups algebra logika sem russian word order problems automata groups jarkko kari nilpotency problem cellular automata siam comput doi ines klimann jean mairesse matthieu picantin implementing computations automaton semi groups implementation application automata lecture notes comput vol springer heidelberg doi ville lukkarila deterministic tiling problem undecidable theoret comput sci doi marvin minsky recursive unsolvability post problem tag topics theory turing machines ann math doi volodymyr nekrashevych groups mathematical surveys monographs vol american mathematical society providence said sidki automorphisms trees growth circuit structure acyclicity math sci new york algebra applications normale paris mathematisches institut address address phortim
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mar threshold estimation stochastic processes small noise yasutaka department applied mathematics waseda university march abstract consider process satisfying stochastic differential equation unknown drift parameter suppose discrete observations given known simple least squares estimator lse consistent numerically unstable sense large standard deviations finite samples noise process jumps propose filter cut large shocks data construct lse data selected filter proposed estimator asymptotically equivalent usual lse whose asymptotic distribution strongly depends noise process however numerical study looked asymptotically normal example filter choosen suitably noise process try justify phenomenon mathematically certain restricted assumptions key words stochastic differential equation semimartingale noise small noise asymptotics drift estimation threshold estimator mighty convergence introduction let stochastic basis stochastic process defined via stochastic integral equation unknown parameter belongs parameter space open bounded convex subset put closure measurable function stochastic process form semimartingale fractional brownian motion hurst paramwhere eter real number satisfying also assume shimizu threshold estimation small noise model probability semimartingale one hence sup well known semimartingale neither converges uniformly semimartingale almost surely moreover suppose decomposition given follows process finite variation martingale typical examples given section suppose process observed discretely time xtnk tnk index omitted notation denote xtnk tnk interest estimate value parameter discrete samples well small noise asymptotics practical advantages small noise asymptotics statistical point view drift estimation samples fixed finite time interval justifiable relatively mild conditions since need observe process long time achieve good estimation drift without small noise assumption usually assume asymptotics terminal time observations goes infinity technical conditions ergodicity uniform moment conditions process need assumed suitable scaling technique regard model small noise model approximately need conditions sometimes difficult check practice see remarks section details computational point view approximation functionals process often available relatively form yoshida pavlyukevich among others well applied finance insurance see takahashi kunitomo takahashi takahashi yoshida uchida yoshida pavlyukevich references therein using small noise model convenient deal applications statistical inference time sampling problems stochastic differential equations small noise well studied many authors theoretical applied point views earlier works models found papers kutoyants laredo developed directions several authors martingale estimating functions studied efficient estimation investigated uchida gloter see also uchida asymptotic expansion approach initiated yoshida see also yasutaka shimizu uchida yoshida among others although works due diffusion noise general noise model also considered recently example long investigate drift estimation driven process see also long long deal inference drift small semimartingale noise paper require noise semimartingale approximately semimartingale sense goal paper statistical inference drift process asymptotics noise vanishes motivation long seen numerical study estimator performed better finite samples long consider case process although extended case semimartingale independent dispersion parameter investigate asymptotic behavior least estimator lse defined lse arg min lse limit show minimum contrast estimator functional mild conditions function see theorems asymptotic distribution generally causes unsatisfactory performance even small enough see numerical results section would due large shocks driving noise easy imagine large jump makes much impact direction drift make drift estimation unstable therefore cutting large jumps could improve performance consider estimator defined follows arg min positive number threshold eliminate large shocks causing bias drift estimation indicator plays role filter split increments large small magnitude shocks see shimizu shimizu yoshida fundamental idea filters would intuitively clear lse asymptotically equivalent however show thing holds true even choosing sequence threshold estimation small noise model carefully moreover see numerical study much better lse furthermore show mighty convergence performance convergence moments stronger result every continuous function polynomial growth asymptotic distribution described asymptotic distribution generally normal unless limiting process wiener process however interesting note sometimes encounter phenomenon seems asymptotically normal numerical study may indicate filtered lse could also asymptotically normal choose different way previous otherwise may observe approximate phenomenon possibly appropriate conditions satisfied although justification phenomena discrete sampling still open problem add discussion points section paper organized follows section prepare notation assumptions present main results discrete samples particular section devoted investigating technical conditions give sufficient conditions noise process section show advantage estimator compared usual lse via numerical study observe asymptotic distribution seems normal finally try give theoretical explanation asymptotic phenomena section proofs main theorems given section main results notation assumptions use following notation process klp given multilinear form rdk vector rdk write ukk note form jth dimension number missing resulting form regarded multilinear form example vector yasutaka shimizu matrix mij inner product quadratic form among others correspondences dimensions clear context also use notation moreover denote forms multilinear form matrix form multilinear form denotes sum squares element often used generic positive constant may differ line line moreover write almost surely denotes space functions times differentiable respect respectively moreover denotes subclass polynomial growth uniformly function write moreover denote asymptotic symbol described unless otherwise noted using notation estimating function given rewritten identity matrix make following assumptions model assumption ordinary differential equation unique solution least one value threshold estimation small noise model positive definite make following conditions limiting process exists sup processes given total variation say quadratic variation satisfy although condition seems hoc give conditions important cases process satisfying see section details asymptotic behavior estimators theorem suppose sequence satisfies remark condition ensures kind negligibility lse see proof makes asymptotically equivalent theorem suppose assumptions theorem dqt square bracket dqt quadratic form dqt matrix case see section theorem suppose assumptions theorem holds true follows every continuous function polynomial growth given theorem yasutaka shimizu examples remarks noise suppose constant brownian motion standard process stability index skewness parameter denoted see cont tankov page notation case limiting variable theorem becomes standard gaussian variable hence estimator asymptotically normal see also long markovian noise let consider case dwt dzt suitable functions pure jump process case reparametrize obtain dwt dzt asymptotically normal uchida verify dwt uniformly fact stochastic integrals continuous respect topology see theorem protter uniform convergence compact sets see theorem jacod shiryaev case use model asymptotic distribution convolution stochastic integrals respect threshold estimation small noise model case reparametrize obtain dqt dqt dwt dzt asymptotic distribution written stochastic integral respect observations consider discretely observed model observed tnn fixed large enough process index estimate drift function sometimes assume parametric model estimate assumption however standard theory parametric inference usually need ergodicity uniform moment conditions supt often restrictive conditions transform model divide sides obtain since stable processes regard weak solution following sde also process suppose small enough given putting reformulate model small enough interpreted small noise sde given model observations reformulate small noise model known constant plotting data interval although ysnk snk snk tnk depends formally use estimator small noise model require restrictive condition yasutaka shimizu conditions let investigate sufficient conditions ensure specified important case process characteristic exponent log exp given measure simple case wiener process follows property stationary independent increments sup eiu sup standard normal distribution function last equality see doob boukai stable process following result proposition symmetric process holds true proof due maximal inequality joulin sup sup also present sufficient conditions process general process let proposition suppose process characteristic exists constant inf lim sup threshold estimation small noise model condition holds true interpret proof let wiener process pure jump process characteristic log exp independent stationary increments property process see sup sup sup note moreover according pruitt follows hence equivalently definition result follows corollary suppose given characteristic eiu note expression proposition consistent index inf proof note assumption yasutaka shimizu let index inf easily see direct computation lim lim inf indeed condition lim inf easier since hence get condition reduced moment conditions measure proposition suppose process characteristic condition holds true proof process following decomposition bwt constants wiener process poisson random measure decomposition given bwt threshold estimation small noise model follows inequality according argument bichteler jacod see also proofs lemma proposition shimizu yoshida see hence similar argument easy see inequality take completes proof numerical study model consider following driven sde standard brownian motion standard symmetric process variance gamma process density obtained brownian subordination gamma process shape scale follows yasutaka shimizu standard brownian motion independent see cont tankov details assume independent sequel set values parameters sample path given figure unbounded variation finite activity jumps infinite compare estimator lse long lse threshold estimator lse blse solution note lse lse lse estimator solution xtn xtn xtn xtn xtn xtn xtn xtn simulations tried results lse given table estimator threshold used respectively simulations iterated times mean standard deviation estimators given tables table values also included discussion numerical results observe estimator improves accuracy estimation sense standard deviation compared usual lse long especially improvement drastic threshold estimation small noise model would parameter stable process much frequent larger jumps order make estimation accurate need make smaller rather larger note case causes large bias reason would asymptotic theory work well small although large enough theory many simulations omitted see least would needed estimation small bias see also case returns better estimation although remains problem choose practice use method proposed shimizu parameters noise process known estimable finally observe normal normalized estimators case see figures according results estimators seem asymptotically normal although theorem necessarily say reference see figures normal lse without filter proposed long figures show usual lse necessarily asymptotically normal theory saying therefore asymptotic phenomena would due filter effects could understand results intuitively follows cutting large jumps process infinite activity jumps remaining small jumps behave brownian motion example suppose driving noise process infinite activity jumps measure put compensated poisson random measure given proof proposition according asmussen rosinski follows equipped certain assumption standard brownian motion therefore appropriate norming estimator limit theorem might integral respect process close brownian motion consideration indicates lse filter approximately asymptotically normal letting converge zero suitable rate great advantage would like make confidence intervals statistical testing next subsection shall try understand phenomenon theoretically could filtered lse asymptotically normal section assume process measure consider following two cases characteristic yasutaka shimizu finite activity case infinite activity case possibly former case would possible show asymptotic normality filtered lse separating increments without jumps shimizu yoshida however known filter enough separate without jumps see shimizu lemma remarks section shall consider following hoc situation understand filtered lse looks like asymptotically normal assumption jumps observed assumption following contrast function make sense constant sup assumption specify define estimator minimum contrast estimator arg min hereafter use following notation measure asymptotic symbols used finite activity case suppose implies written poisson process intensity sequence distribution special case following result taking faster speed magnitude jumps threshold estimation small noise model theorem suppose given hold true moreover suppose hence asymptotically normal dwt infinite activity case theorem let process suppose holds true moreover suppose log exists constant log furthermore suppose exists brownian motion independent following weak convergence holds true dbt hence asymptotically normal shall give concrete example satisfies situation remark assumption approximate component compensated small jumps wiener process see theorem asmussen rosinski according proposition simple condition sufficient moreover note requires high excludes cases compound poison process gamma process see examples therefore assumed yasutaka shimizu remark although result ideal situation jumps observable imagine phenomenon approximately occurs simulations presented figures sufficiently small filter successfully cut jumps whose sizes larger however would hard show similar result observations completely discrete original setting filter exactly exclude increment includes jumps whose sizes larger see also shimizu filter infinite activity cases complete analysis discretely observed cases would important work future lse lse true lse lse true lse lse true table results lse without filter based long find standard deviation large especially implies unstability estimation would like improve stability using filter proofs preliminary lemmas shall first establish preliminary lemmas show main theorems later threshold estimation small noise model normal plot sample quantiles sample quantiles normal plot theoretical quantiles theoretical quantiles lse figure normal left right results show right tail especially heavier normal distribution normal plot sample quantiles sample quantiles normal plot theoretical quantiles theoretical quantiles lse figure normal left right results show tails heavier normal distribution yasutaka shimizu true true true table results filter compared lse improvement drastic although one less make estimation accurate need make smaller values seems enough meet asymptotic conditions must tend threshold estimation small noise model true true true table results filter smaller well however careful observing case estimators negatively biased would small meet corresponding asymptotic condition yasutaka shimizu normal plot sample quantiles sample quantiles normal plot theoretical quantiles figure normal distribution theoretical quantiles left right small seems almost gaussian normal plot sample quantiles sample quantiles normal plot theoretical quantiles theoretical quantiles figure normal left right small distribution seems almost gaussian threshold estimation small noise model time figure sample path model evaluate functions discrete samples use following notation long discretized process ytn stands integer part since semimartingale consider decomposition process finite variation martingale stopping time localization inf inf convention inf hence note almost surely lemma holds addition suppose holds yasutaka shimizu proof follows small enough hence take argument proof lemma long obtain since bounded probability hence fact yields proof moreover note sup sup sup noticing linear growthness function inequalities sup sup sup used inequality last inequality completes proof form proof following corollary obvious threshold estimation small noise model corollary suppose holds sup addition sup lemma follows lim uniformly proof noticing argument proof lemma long sup sup ect therefore consequence lemma let suppose uniformly addition suppose holds sup sup proof since sup sup yasutaka shimizu sup ytn first term side inequality holds sup ytn sup ysn ysn duds ysn sup sup sup ytn lemma hence proof ends show second term tends zero probability let show implies lemma jacod first show note since follows show lemma suffices note holds sup sup hence follows assumption large enough sup threshold estimation small noise model ftn sup ftn since bounded probability proof proof similar ends proof proof easy estimates corollary since assuming omit details proof completed next lemma version toeplitz lemma see also shimizu need result proof next lemma lemma let ank positive bounded sequence put ank suppose sequence xnk satisfies following conditions sup fixed lim sup sequence ani xni lemma let holds dqt proof since assuming uniform convergence dqt ytn ytn dmt ytn dqt yasutaka shimizu suppose simplicity notation putting ank xnk qtk since depends qtk qtk ank xnk assumption moreover sum itnis clear also converges zero probability one using lemma indeed convergence clear lim sup lim moreover ank hence see toeplitz lemma ank xnk probability one therefore probability one follows lemma sup sup sup ytn using markov inequalities dmt dmt ytn ytn definition integrand last term bounded hence taking limit see dominated convergence theorem completes proof threshold estimation small noise model lemma let assume uniformly addition assume dim sup proof proof similar one lemma long slight extension semimartingale version see also remark clear proof indicator essential proof omitted note sup sup ysn ysn sup ysn ysn sup ysn dqt sup ysn dqt assumption condition ysn hence assumption corollary yields already shown proof lemma hence proof ends show noticing bounded convex set admits following sobolev inequality sup klq klq yasutaka shimizu dim see sup ysn ysn using inequality ysn completes proof lemma let assume proof using tnk easily find converges zero probability argument lemma long follows lemma hence proof completed threshold estimation small noise model proof theorem lse given let shall show asymptotically equivalent lse respectively given minimum contrast estimators contrast functions argument proof theorem remark long need show sup therefore since sup sup sup sup sup sup last first second terms converges zero probability lemmas completes proof proof theorem use following notation follows taylor formula yasutaka shimizu let show dqt sup result follows argument proof theorem long see also uchida follows show last first term converges zero probability evaluation proof lemma second term converges dqt probability lemma third term goes zero probability lemma similarly follows expression proof lemma long hence lemmas yield using facts consistency result theorem show completely argument proof theorem long therefore proof completed threshold estimation small noise model proof theorem denote since theorem proof ends show supn proof let define random fields exp since see sup setting consider following condition random fields every sup called polynomial type large deviation inequality pldi investigated yoshida details pldi holds true sup sup sup sup used first inequality therefore proof ends show sufficient conditions found paper yoshida shall verify conditions given theorem see also ogihara yoshida masuda simplified descriptions conditions applying taylor formula notation given proof theorem log dtds yasutaka shimizu means could partially locally asymptotically quadratic plaq starting point according theorem take tuning parameters given pldi holds true following satisfied use conditioning numbers make correspondences clear every sup sup moreover given small enough sup small enough sup sup sup given proof theorem matrix deterministic positive definite exists deterministic positive number note notational correspondence easily check conditions true lemma also true lemma moreover conditions clear assumptions respectively proof ends show hence proof completed proof theorem first shall show consistency threshold estimation small noise model since suppose jumps specified following negligibility obtained sup since following lemma type results lemmas lemma let suppose uniformly proof lemma obvious modification proofs lemmas using negligibility condition argument proof theorem consistency follows next note proof theorem show dwt obtain consequence convergence sup yasutaka shimizu holds true due lemma argument proof lemma note note argument proof lemma assumption moreover easy see furthermore note observe measurability tnk independent therefore trace result lemma yields dwt completes proof threshold estimation small noise model proof theorem note asymptotic conditions follows different situation previous theorem setting show following lemma lemma let suppose log follows uniformly proof lemma note sup sup sup second term side inequality converges zero probability lemma long last first term rewritten immediately see since independent yasutaka shimizu moreover log assumption hence every lemma jacod pto prove uniformity convergence show tightness sequence shall use theorem appendix ibragimov minskii shall show using independent property sup therefore bounded inequality similarly proved since hence proved finally shall show note sup sup sup threshold estimation small noise model follows assumption given lemma sup ysn ysn sup ytn sup sup since lemma follows sup sup tnk sup since moreover noticing sup obtain completes proof putting using lemma easily show sup yasutaka shimizu identifiability condition yield consistency suppose process form cwt wiener process poisson random measure note associated jumps hereafter put using notation previous theorem proof ends show dbt note since independent trace threshold estimation small noise model last equality due therefore integer see due condition hence argument proof lemma obtain following inequality sup using estimates proof lemma estimates see let thanks theorem asmussen rosinski follows assumption exists wiener process independent moreover since lemma joint convergence hence follows theorem jacod shiryaev ytn yasutaka shimizu dbt completes proof statement proved acknowledgement author would like thank anonymous referees valuable suggestions proposals significantly improve manuscript research partially supported jsps kakenhi scientific research grant number references asmussen rosinski approximations small jumps processes view towards simulation appl bichteler jacod calcul malliavin pour les diffusions avec sauts existence une densite dans cas unidimensionnel lecture notes math berlin boukai explicit expression distribution supremum brownian motion change point comm statist theory methods cont tankov financial modelling jump processes chapman boca raton doob heuristic approach theorems ann math statistics maximum contrast estimation diffusion processes discrete observations statistics jacod estimation diffusion coefficient multidimensional diffusion process ann inst henri prob statist gloter estimation stochastic differential equations small diffusion coefficient stochastic process ibragimov minskii statistical estimation berlin threshold estimation small noise model jacod shiryaev limit theorems stochastic processes second edition berlin joulin maximal inequalities stable stochastic integrals potential kunitomo takahashi asymptotic expansion approach valuation interest rate contingent claims math finance kutoyants parameter estimation stochastic processes heldermann berlin kutoyants identification dynamical systems small noise kluwer dordrecht laredo sufficient condition asymptotic sufficiency incomplete observations diffusion process ann long least squares estimator discretely observed processes small noises statist prob letters long parameter estimation class stochastic differential equations driven small stable noises discrete observations acta mathematica scientia note least squares estimator discretely observed ornsteinuhlenbeck processes small noises statist prob letters masuda convergence gaussian random fields ergodic levy driven sde observed high frequency annals statistics long shimizu sun least squares estimators discretely observed stochastic processes driven small noises multivariate analysis ogihara yoshida analysis stochastic differential equation jumps stat inference stoch pavlyukevich ruin probabilities small noise heavy tails appl stoch models bus ind pavlyukevich first exit times solutions stochastic differential equations driven multiplicative noise heavy tails stoch protter stochastic integration differential equations second edition berlin yasutaka shimizu pruitt growth random walks processes ann probability shimizu discretely observed ergodic diffusion processes infinitely many jumps statist infer stochastic shimizu threshold selection filter discretely observed jump processes statist methods shimizu local asymptotic mixed normality discretely observed nonrecurrent processes ann inst statist shimizu yoshida estimation parameters diffusion processes jumps discrete observations statist infer stochastic small dispersion asymptotics diffusion martingale estimating functions preprint department statistics operation research university copenhagen copenhagen uchida small diffusion asymptotics discretely sampled stochastic differential equations bernoulli takahashi asymptotic expansion approach pricing contingent claims financial markets takahashi yoshida asymptotic expansion scheme optimal investment problems stat inference stoch uchida estimation discretely observed small diffusions based approximate martingale estimating functions scand statist uchida approximate martingale estimating functions stochastic differential equations small noises stochastic process uchida yoshida asymptotic expansion small diffusions applied option pricing stat inference stoch yoshida asymptotic expansion maximum likelihood estimators small diffusions via theory probab theory relat fields yoshida asymptotic expansion statistics related small diffusions japan statist yoshida polynomial type large deviation inequalities analysis stochastic differential equations ann inst statist
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approximating partition functions constant time feb vishesh frederic elchanan abstract study approximations partition function dense graphical models partition functions graphical models play fundamental role statistical physics statistics machine learning two main methods approximating partition function markov chain monte carlo variational methods impressive body work mathematics physics theoretical computer science provides conditions markov chain monte carlo methods converge polynomial time methods often lead polynomial time approximation algorithms partition function cases underlying model exhibits correlation decay theoretical guarantees performance variational methods one exception recent results risteski considered dense graphical models showed using variational methods possible find additive approximation log partition function time even regime correlation decay hold show essentially conditions additive approximation log partition function found constant time independent particular results cover dense ising potts models well dense graphical models interaction also apply low threshold rank models best knowledge results first give constant time approximation log partition functions first use algorithmic regularity lemma estimating partition functions application results derive constant time algorithm approximating magnetization ising potts model dense graphs massachusetts institute technology department mathematics email visheshj massachusetts institute technology department mathematics email fkoehler massachusetts institute technology department mathematics idss supported onr grant nsf email elmos introduction one major algorithmic tasks areas markov chain monte carlo statistical inference machine learning approximating partition functions graphical models partition function approximate partition function used computing marginals posteriors two basic inference tasks graphical models moreover intimate connection computing partition function markov chain monte carlo methods particular jerrum sinclair showed possible approximate partition function models rapidly mixing markov chain exists hand models approximation partition function results rapidly mixing chain key results theory mcmc provide conditions existence rapidly mixing chains therefore allow efficient approximations partition functions follow work different line work based fact measure associated graphical model characterized variational problem thus approximating partition function achieved solving optimization problems recently general guarantees performance variational methods one exception results risteski showed using variational methods possible find additive approximation log partition function dense graphical models time even interesting regime correlation decay hold denotes total weight interactions graph guaranteed priori upper bound log partition function following risteski consider dense graphical models show possible find additive approximation log partition function randomized constant time thus main result stated informally theorem dense graphical models possible compute log partition function additive error time depends density parameter note approximation guarantee theorem rather weak inference tasks one interested good approximation partition function results like provide approximation partition function within exp multiplicative factor approximation useful inference marginalization tasks note however easy see constant time impossible get approximation partition function see theorem formal statement features graphical models reflected log partition function particular show theorem utilize results obtain magnetization ising models constant time similar statements hold dense graphical models well exact statement theorem provides magnetization model parameters close model interested show necessary constant time impossible break symmetry phases note results hold general markov random fields hand even case ferromagnetic ising models polynomial time guarantees previously provided see recent work also note results thought generalizing classical results approximability similar problems dense graphs fact problem special case computing log partition function antiferromagnetic ising model entries negative graph equal edge weights limit kjk entropy term becomes negligible note case function consider choosing according independent rademacher random variables approximation actually gives ptas first ptas dense graphs given arora karger karpinski ran time improved frieze kannan using weak regularity lemma remains essentially fastest runtime known algorithm matches runtime recent works shown sample complexity problem number entries adjacency matrix need probed case determining correct sample complexity remains interesting open problem see section also interesting note optimization case algorithms based regularity algorithms based convex programming hierarchies see dependence result also dependence showing phenomena extends problems well suggesting difficult beat runtime overview results notation definitions consider ising models spins valued interactions ferromagnetic varying strengths simplicity primarily restrict case external fields though clear methods extend case well remark definition ising model probability distribution discrete cube form exp exp collection pnentries arbitrary real symmetric matrix exp normalizing constant called partition function ising model often need consider matrix vector exactly arrange entries single vector matter denote clarity resulting vector definition vector norm denoted defined also define vector norms give rise matrix norms viewing matrix vector use sequel cases particular importance another matrix norm need arises viewing matrix linear operator via matrix multiplication definition matrix define following focus ising models definition ising model combinatorial perspective models generalization dense graph models statistical physics perspective generalize complete graph models curieweiss model model main results provide algorithms run constant time order provide guarantees problems unbounded input size work usual assumptions computational model algorithms thus probe matrix entry time also note standard chernoff bounds follows set vertices test membership time also estimate additive error constant time using samples approximation always suffice sake clarity exposition henceforth ignore technical detail assume access main results main theorem ising models theorem exists universal constant ising model nodes algorithm running time log computes additive approximation log probability least methods extend straightforward manner higher order markov random fields general finite alphabets well review definitions section simplicity state result case binary markov random fields theorem fix exists universal constant algorithm binary random field running time log computes additive approximation log probability least previous theorem possible improve dependence expense introducing factor running time theorem fix exists universal constant binary markov random field algorithm additive approximation log running time log computes kjk probability least remark although simplicity stated theorem theorem markov random fields immediately seen results extend general markov random fields order simply applying theorem theorem see section definitions particular directly handles dense ising models external fields remark error term theorem two terms kjk log exp therefore log log similar statements hold theorems well thus interesting applications theorem may assume wlog dominant term dominant mean least small fraction term also able handle case ising models whose interaction matrices low threshold ranks although constant time simplicity record result regular low threshold rank models definitions presented section following remark assume theorem exists universal constant regular ising model nodes rank model algorithm running time poly computes additive approximation log approximating expected total magnetization given ising model exp one fundamental questions one ask many spins expectation case ferromagnetic ising model corresponds strongly system magnetized accordingly define expected total magnetization ising model expectation respect ising distribution perhaps surprisingly results easily show dense ising models one obtain sense approximation expected magnetization constant time precisely theorem consider ising model exp let exp let denote expected total magnetization prh find additive approximation constant time depending kjk proof well known one express moments spin systems termspof derivatives log partition function particular ising model exp consider family perturbed ising models defined prh exp log exp exp denotes expectation respect ising distribution perturbed particular log equals expected total magnetization ising model started moreover since jensen inequality log exp see log convex particular log log log log log finally mean equation given value theorem lhs setting theorem remark evaluate lhs rhs desired error constant time remark unfortunately impossible approximate constant time magnetization specified value external fields example consider ising model vertices large otherwise let set except set uniformly chosen uniformly chosen note dense ising model per definition note also nodes ising model complete graph one random node external field easy see magnetization fact large constant implies conditioning one vertex taking value results dramatic change magnetization vertices particular magnetization order order thus follows need queries order determine magnetization case note example corresponds phase transition particular every values see general references ising model complete graph results computing magnetization readily extend models example potts models compute color expected number nodes color error close external field similarly easy check compute pother statistics accuracy instance ising model approximate tightness results mentioned introduction results qualitatively tight make precise proving following lower bound accuracy algorithms additively approximating log theorem fix possibly randomized algorithm probes entries returning estimate log exists input instance makes error least probability least proof prove claim reduction hypothesis testing problem specifically show functions exist two different dense ising models apart algorithm makes probes least distinguish two probability greater immediately implies algorithm estimate log least one two inputs must make error probability least given input otherwise could use least output distinguish two models better probability checking log output closer let instance size taken sufficiently large consider two ferromagnetic ising models defined follows underlying graph complete graph vertices many edges randomly selected weight remaining many edges assigned weight note since model indeed sufficiently large underlying graph complete graph vertices edges weight respectively easily seen denote partition functions models log limm limm logmzm limm limm therefore sufficiently large log log log log probability greater show algorithm distinguish input algorithm probes fix split since randomized algorithm viewed mixture deterministic algorithms must exist deterministic algorithm success probability distinguishing least large let first edge queried let next edge queried assuming define similarly without loss generality algorithm uses available queries let event juk see equal event always happens input thus total variation distance observed lemma know distribution fails probability least therefore see fails probability least proves result note input instances given theorem algorithm use approximation models probability finding queries output optimal dependence running time parameters remains interesting open problem outline techniques illustrate main reason intractability log partition function ising model consider ferromagnetic case exp case clear given magnetized state almost spins either likely given unmagnetized state spins almost evenly split see however since number states exactly spins equal simply total number strongly magnetized states exponentially smaller total number unmagnetized states therefore given unmagnetized state less likely given magnetized state may well case total probability system unmagnetized state greater system magnetized state paper present approach dealing tradeoff dense ising models theorem based algorithmic regularity lemma frieze kannan theorem roughly speaking lemma allows efficiently partition underlying weighted graph small number blocks manner quantities associated graph approximately depend numbers edges various blocks lemma show log partition function fits framework similar statement may viewed limiting case model antiferromagnetic interactions allowed tends infinity first obtained key point regularity lemma number blocks depends desired quality approximation size underlying graph together previous statement shows lemma one approximately rewrite sum computing partition function terms polynomially many nonnegative summands opposed exponentially many nonnegative summands started provides way resolve tradeoff log largest summand much smaller sum approximates log partition function well lemma preceding discussion reduces problem estimating log partition function optimization problem although different one however stated problem solve constant time lemma show granulate parameters reduce problem constant size algorithm solves problem efficiently via convex programming proofs theorem theorem theorem follow similar outline application theorem replaced theorem theorem theorem respectively preliminaries case dense ising model reduction problem solved constant time based algorithmic weak regularity lemma frieze kannan stating introduce terminology throughout section deal matrices whose entries index definition given define cut matrix otherwise definition cut decomposition expresses matrix say cut decomposition width coefficient length error ready state algorithmic weak regularity lemma frieze kannan theorem let arbitrary real matrix let time find cut decomposition width coefficient length probability error reducing number summands using weak regularity next lemma shows purpose additively approximating function plog partition may well work matrix define exp lemma let matrix interaction strengths ising model let cut decomposition theorem log log proof note therefore exp taking first sum inequalities log get log exp finally noting proof follows recall cut matrix vertex sets pass common refinement gives disjoint sets every one union atoms terms define another approximation partition function exp ranges elements log log binary entropy function note represents net spin similarly next lemma shows approximating log suffices obtain approximation log combined lemma therefore see approximating log sufficient approximating log lemma log log log proof first observe thus letting see exp exp summation terms possible values take get exp ranges elements similarly next since follows exp log finally apply stirling formula log get desired result approximating reduced sum constant time using convex programming far seen problem estimating log partition function reduced problem approximating log next simple lemma reduces estimation log optimization problem lemma let arg maxy exp log log proof follows immediately noting sum many nonnegative summands least large largest summand larger times largest summand following lemma shows estimating contribution term corresponding vector suffices know components constant precision reduces optimization problem one constant size lemma let denote matrix interaction strengths ising model let cut decomposition theorem given get proof theorem know follows since completes proof present algorithm approximates log largest summand iterating possible proportions block proportion approximately solves following convex program parameter specified max infeasibility program means guess proportion various combinatorially possible recall may significant overlap various hand program feasible actually obtain approximate maximum entropy configuration roughly prescribed number calculating contribution configuration maximizing contribution iterations gives good maximum summand hence content next lemma log additive approximation lemma algorithm log proof lemma suffices prove claim log replaced defined statement lemma let correspond cell falls values corresponding slightly better approximation log found estimating discrete sum region integral using lemma bound error approximating integral via random walk however algorithm slower algorithm turns gain accuracy negligible application algorithm convex programming method estimate log partition unction let either find solution verify infeasibility ellipsoid method feasible let denote maximum value let let exp nhr max end end return log since result solving additive error since feasible region convex program definition follows nhr sup ranges feasible region combining lemma gives log nhr hand know let denote point optimum convex program attained define otherwise similar inequality get log log log log defined algorithm terms second inequality used lemma triangle inequality last line used lemma bound last term since definition since theorem get provided finishes proof lemma let natural number let let log proof observe immediate since entropy concave maximized thereby implying remaining case opposite sides definition know hence mean value theorem thus log log already log log putting together ready prove theorem proof theorem first show get algorithm constant success probability boost desired probability success using standard median technique constant success algorithm algorithm weakly regular partition generated theorem regularity parameter correctness algorithm note output algorithm log log log log log log log log log first bound lemma second bound lemma last bound lemma log take sufficiently large runtime computing weakly regular partition success probability theorem runtime algorithm many points solving convex program well deciding feasibility done time poly log log standard guarantees ellipsoid method finally boost success probability use standard median trick repeat algorithm log times independently take median outputs works easily proven standard chernoff bounds extensions general markov random fields simplicity restrict markov random fields binary alphabet readily seen techniques extend markov random fields general finite alphabets well definition binary markov random field order probability distribution discrete cube form exp xik arbitrary collection real numbers note sum map xik may also viewed eik denotes standard basis vector view subset normalizing constant exp called partition function markov random field moreover entries vanish say markov random field definition binary markov random field order indicated earlier approach general markov random fields mirrors approach ising model essentially one similar algorithmic regularity lemma general matrices definition matrix map definition real number define matrix otherwise theorem suppose arbitrary matrix assume fixed let let time probability least find cut decomposition width coefficient length error case error cut decomposition refers view matrix linear operator via formula given theorem proof proceeds exactly argument lemma shows replace possibly incurring additive error made replacement similar optimization scheme lets deduce theorem proof theorem identical difference obtain cut decomposition smaller width using following theorem theorem let arbitrary matrix assume fixed let let time probability least find cut decomposition width coefficient length error ising models low threshold rank also consider ising models low threshold rank simplicity consider regular case noting results generalise appropriate modifications case well definition regular weighted ising model one normalized adjacency matrix regular ising model matrix entries denote matrix definition squares threshold rank regular ising model defined denote eigenvalues note since eigenvalues absolute value criterion low threshold rank general defining number eigenvalues absolute value strictly greater definition differs slightly definition author defined denote eigenvalues matrix abs defined abs definition definition related via standard linear algebra follows fact since linear algebra fact shows low rank setting strictly generalises setting regular ising model used regular case methods extend straightforwardly low threshold rank setting due following algorithmic regularity lemma gharan trevisan theorem let matrix interaction strengths regular ising model let let time poly find cut decomposition width error recall case ising model bounds cut decomposition actually used width error quite similar since hence exactly analysis case conclude theorem omit details acknowledgements thank david gamarnik insightful comments andrej risteski helpful discussions related work yufei zhao introducing reference references noga alon fernandez vega ravi kannan marek karpinski random sampling approximation comput system sanjeev arora david karger marek karpinski polynomial time approximation schemes dense instances problems proceedings annual acm symposium theory computing stoc pages new york usa acm alexander barvinok combinatorics complexity partition functions algorithms combinatorics david blei alp kucukelbir jon mcauliffe variational inference review statisticians arxiv preprint wenceslas fernandez vega claire linear programming relaxations maxcut proceedings eighteenth annual symposium discrete algorithms pages society industrial applied mathematics richard ellis entropy large deviations statistical mechanics springer alan frieze ravi kannan regularity lemma approximation schemes dense problems proceedings annual ieee symposium foundations computer science pages alan frieze ravi kannan quick approximation matrices applications combinatorica shayan oveis gharan luca trevisan new regularity lemma faster approximation algorithms low threshold rank graphs approximation randomization combinatorial optimization algorithms techniques pages springer martin alexander schrijver geometric algorithms combinatorial optimization volume springer science business media piotr indyk sublinear time algorithms metric space problems proceedings thirtyfirst annual acm symposium theory computing pages acm jerrum sinclair approximating permanent siam jerrum sinclair approximation algorithms ising model extended abstract automata languages programming pages jerrum sinclair vigoda approximation algorithm permanent matrix entries journal acm jingcheng liu alistair sinclair piyush srivastava ising partition function zeros deterministic approximation focs santosh vempala fast algorithms logconcave functions sampling rounding integration optimization foundations computer science focs annual ieee symposium pages ieee claire mathieu warren schudy yet another algorithm dense max cut greedy proceedings nineteenth annual symposium discrete algorithms soda pages philadelphia usa society industrial applied mathematics andrej risteski calculate partition functions using convex programming hierarchies provable bounds variational methods colt david sherrington scott kirkpatrick solvable model physical review letters alistair sinclair mark jerrum approximate counting uniform generation rapidly mixing markov chains information computation martin wainwright michael jordan graphical models exponential families variational inference foundations trends machine learning jonathan yedidia william freeman yair weiss understanding belief propagation generalizations yuichi yoshida yuan zhou approximation schemes via hierarchy dense constraint satisfaction problems assignment problems proceedings conference innovations theoretical computer science pages acm figure available jpg format http
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submitted annals probability arxiv exponentially concave functions new information geometry may soumik leonard university university southern function exponentially concave exponential concave consider exponentially concave functions unit simplex previous paper showed gradient maps exponentially concave functions provide solutions optimal transport problem give better gradient approximation ordinary concave functions approximation error called different usual bregman divergence using tools information geometry optimal transport show induces new information geometry simplex consisting riemannian metric pair dually coupled affine connections defines two kinds geodesics show induced geometry dually projectively flat flat nevertheless prove analogue celebrated generalized pythagorean theorem classical information geometry hand consider displacement interpolation lagrangian integral action consistent optimal transport problem show action minimizing curves dual geodesics pythagorean theorem also shown interesting application determining optimal trading frequency stochastic portfolio theory introduction definition exponential concavity let convex say function exponentially concave concave convention set throughout paper let open unit simplex regarded collection strictly positive probability distributions set elements due applications mind research partially supported nsf grants msc subject classifications primary secondary keywords phrases information geometry optimal transport exponential concavity generalized pythagorean theorem functionally generated portfolio stochastic portfolio theory pal wong although many generalizations possible interesting property exponentially concave functions gradient maps give better firstorder approximation ordinary concave functions introduced concept let differentiable exponentially concave function concavity implies log euclidean gradient clearly approximation sharper linear approximation error approximation log extra concavity exponentially concave functions found several recent applications analysis probability optimization example equivalence entropic conditions bochner inequality metric measure spaces established using notion convexity negative convex function exponentially concave better gradient approximation also led better algorithms optimization machine learning although authors tend replace logarithmic term quadratic approximation one primary applications mind related stochastic portfolio theory author considers gradient map exponentially concave function map closure following restatement found proposition let differentiable exponentially concave function define ith standard basis shown keeping standard definitions subject call map portfolio map vein also see articles distinguished bregman divergence defined bregman divergence introduced widely applied statistics optimization see difference consider two fundamental examples exponentially concave functions divergence also known relative entropy given log shown relative entropy bregman divergence shannon entropy log hand fix consider cross entropy log exponentially concave function whose associated portfolio map constant corresponding given log log quantity sometimes referred free energy statistical physics finance called diversification return excess growth rate rebalancing premium volatility return introduced optimal transport problem solved using exponentially concave functions unit simplex cost function defined exi log strictly convex recall details transport problem section relationship exponentially concave functions suffices say given pair borel probability measures optimal coupling two respect cost expressed terms portfolio map exponentially concave function simplex related cost function appears completely different context finding polytopes given geometric data also appears related study moment measures introduced see page particular contributions paper show information geometry provides elegant geometric structure underlying exponential concavity ldivergence optimal transport problem motivating question starting point work suppose exponentially concave function associated geometrically characterize triplets pal wong primal geodesic dual geodesic fig generalized pythagorean theorem answer question determines optimal frequency rebalancing portfolio generated see section also see section transport interpretation inequality using tools information geometry show exponentially concave functions induce new geometric structure simplex regarded smooth manifold probability distributions let exponentially concave function require smooth euclidean hessian strictly positive definite everywhere see assumption induced geometric structure consists riemannian metric dual pair affine connections connections define via parallel transports two kinds geodesic curves called primal dual geodesics interestingly duality geometry goes hand hand duality related optimal transport problem work first exploits connection summarize main results follows first give answer motivating question theorem generalized pythagorean theorem given consider dual geodesic joining primal geodesic joining consider riemannian angle geodesics see proposition expresses riemannian metric normalized euclidean hessian difference positive zero negative depending whether angle less equal greater degrees see figure also prove remarkable properties geodesics exist exponentially concave functions explicit coordinate systems primal dual geodesics time changes euclidean straight lines theorem words new geometry dually projectively flat particular primal geodesics euclidean straight lines time reparameterization moreover primal dual connections constant sectional curvature respect riemannian metric thus satisfy einstein condition corollary primal dual geodesics also constructed time changes riemannian gradient flows functions theorem remarkable geodesic equations depend local properties near gradient flows global involve derivatives indeed relation known limited families divergence including bregman divergence shown chapter generalized pythagorean theorem holds bregman divergence induces dually flat geometry prove resulting geometries flat theorem extensions generalized pythagorean theorem hold certain spaces see example theorem involve extra terms best knowledge theorem first exact pythagorean theorem holds geometry dually flat difference also given optimal transport interpretation section extend static transport problem transport problem corresponding convex lagrangian action theorem show action minimizing curves reparametrized dual geodesics addition satisfy intermediate time optimality condition allows consistent displacement interpolation formulation probability measures unit simplex previously studies focused almost exclusively wasserstein spaces corresponding cost functions metric polish space suitable properties displacement interpolation related concept displacement convexity introduced thesis ideas grown immensely important classical wasserstein transport fundamental implications geometry physics probability pde see chapter thorough discussion lagrangian although convex superlinear therefore covered standard theory however expect lead many equally remarkable properties results suggest plenty problems research generalizing theorem three points interest stochastic pal wong lio theory displacement interpolation become extremely important topic optimal transport theory extensions riemanninan manifolds done led new functional inequalities another vein defines ricci curvatures metric measure spaces terms displacement interpolation displacement convexity expect displacement interpolation paper lead new otto calculus chapter related pdes equations appears bregman divergence two entire family divergences special properties corresponding optimal transport problems example see extends optimal transport problem via cumulant generating function general probability distribution also believe new information geometry useful dynamic optimization problems objective function multiplicative time finally naturally interest study exponential concavity general convex domains related literature mentioned bregman divergence general divergence set usually manifold probability distributions function divergences metrics general since may asymmetric may satisfy triangle inequality apart bregman divergence many families divergences applied information theory statistics areas see survey catalog divergences among divergences bregman divergence plays special role induces dually flat geometry underlying space first studied context exponential families statistical inference gave rise information geometry geometric study manifolds probability distributions furthermore bregman divergence enjoys properties generalized pythagorean theorem projection theorem led numerous applications see introductions beautiful theory related concept dual affine connection also useful affine differential geometry see dually projectively flat manifolds characterized terms bartlett tensors conformal flatness identify new important class examples show concrete applications work motivated study mathematical finance recently optimal transport applied financial problems robust asset pricing see example line work somewhat different flavor although share goal development exponentially concave functions free mathematical finance portfolios generated exponentially concave functions generate profit due fluctuations sequence representing stock market idea sometimes called volatility harvesting leads naturally transport problem shown philosophy work interpreted developing notion modelfree volatility outline paper next section recall optimal transport problem formulated using exponential coordinate system relation functionally generated portfolio also reviewed section relate exponential concavity give transportmotivated definition duality plays crucial role reviewing basic concepts information geometry derive section geometric structure induced exponentially concave function properties new geometry studied section particular characterize primal dual geodesics prove generalized pythagorean theorem interesting application mathematical finance finally section apply geometric structure construct displacement interpolation associated optimal transport problem technical computational details gathered appendix optimal transport portfolio maps section recall optimal transport problem using exponential coordinate system also review definition functionally generated portfolio explain relates transport problem exponential coordinate system exponential coordinate system defines global coordinate system regarded smooth manifold section definition exponential coordinate system exponential coordinate given log denote map convention set inverse transformation given log log defined pal wong exponential coordinate system first several coordinate systems introduce simplex changing coordinate systems function expressed function vice versa explicitly function expressed exponential coordinates simplify notations simply write depor pending coordinate system used example log cross entropy transport problem refer reader books introductions optimal transport interplay analysis probability geometry let equipped standard euclidean metric topology let borel probability measures respectively coupling mean borel probability measure whose marginals respectively let set couplings set always contains product measure given consider optimal transport problem cost defined inf expectation taken probability measure random element distribution optimal coupling takes form measurable map say monge transport map general may consider optimal transport problem replaced general cost function denoted general polish spaces classical example power underlying metric especially costs rich delicate theories developed euclidean spaces riemannian manifolds geodesic metric measure spaces however consider cost function defined remark cost function log differs linear term plays role optimal transport thus may consider instead advantage jensen inequality equals zero consistent notations use cost function paper exponentially concave functions definition monotonicity subset monotone satisfies following property finite collection permutation set well known monotonicity mild technical conditions necessary sufficient solution criteria general optimal transport problem see chapter particular coupling optimal support monotone functionally generated portfolio point convenient introduce concept functionally generated portfolio although possible present theory without reference concepts stress portfolio map gives additional structure transport problem found cases also main examples theory well key quantities induced riemannian metric best expressed terms portfolios mathematically portfolio regarded normalized gradient section apply information geometry functionally generated portfolios functionally generated portfolio introduced following refined definition taken definition functionally generated portfolio portfolio map mean function let exponentially concave say portfolio map generated call log generating function positive concave generating function known unique given additive constant differentiable necessarily given throughout paper impose following regularity conditions exponentially concave function assumption regularity conditions pal wong function smooth infinitely differentiable euclidean hessian strictly negative definite everywhere particular strictly concave moreover shown function defined maps let discuss conditions briefly differentiability needed define differential geometric structures terms derivatives ldivergence theory requires three times continuously differentiable convenience simply assume smooth strict concavity guarantees strict positive definiteness hessian implies induced riemannian metric henceforth let exponentially concave function satisfying assumption let given portfolio map generated cost function always refers one defined general cost function denoted using shown expressed form log give several examples functionally generated portfolios log generating functions many examples found chapter particular portfolios play special role taken basic example theory example examples functionally generated portfolios market portfolio identity map generated constant function assumption hold portfolio constant map generated log special case called portfolio iii portfolio let fixed parameter consider portfolio map defined shown generating function log exponentially concave functions convex combinations known set functionally generated portfolios convex indeed let generated generated portfolio map generated erating function geometric mean fact used formulate study nonparametric estimation functionally generated portfolio following result taken proposition portfolio map following statements equivalent exists exponentially concave function generates sense portfolio map multiplicatively cyclical monotone mcm following sense sequence satisfying iii define map log regarded function exponential coordinate words define way exponential coordinate graph map monotone using result showed optimal transport problem solved terms functionally generated portfolios simple interesting explicit example direct generalization case treated section example product gaussian distributions transport problem let product gaussian distributions pal wong also let optimal transport map measures given map portfolio map following variant portfolio discussed example iii log coefficients chosen log wwni optimal transport duality duality make use notion optimal transport theory definitions use standard found chapter refers cost function also recall underlying spaces variables respectively define inf similarly function defined inf say exists similar functions function defined define set singleton call write similar definitions hold function exponentially concave functions let definition every pair equality holds generalized version fenchel identity see section used frequently paper first lemma relates exponential concavity note cost function asymmetric equivalent change variable lemma exponential concavity following statements equivalent exponentially concave function defined iii function defined exponential coordinate proof prove implication others proved similarly suppose holds consider concave function theorem extend continuously closure extend affine hull setting extended function closed concave function convex duality see theorem exists family affine functions inf since replacing sequence may assume without loss generality strictly positive parameterize form positive constants note extra constant term required since writing pal wong log aani switching exponential coordinates log log log log log log log log follows inf log define setting inf log log infimum empty set inf shows following analogue classical legendre transformation proof standard lengthy given appendix theorem transformation let exponentially concave function satisfying assumption let defined portfolio map generated given consider function defined via exponential coordinate system given log moreover map injective exponentially concave functions primal euclidean dual euclidean primal exponential dual exponential fig coordinate systems let range given fact map diffeomorphism whose inverse also function smooth open set although general strict subset theorem dual variable defines global coordinate system manifold theorem use another coordinate system called dual euclidean coordinate system thus four coordinate systems euclidean primal dual dual euclidean see definition following frequently switch coordinate systems facilitate computations avoid confusions let state conventions used let given definition coordinate systems call identity map unit simplex defined primal euclidean coordinate system range let log log primal exponential coordinate system range pal wong dual exponential coordinate system range dual euclidean coordinate system defined composition see figure illustration always represent point particular unless otherwise specified dual sense convention let notation switching coordinate systems identify spaces using coordinate systems definition function one spaces write depending coordinate system used also record useful fact formula analogous first statement derived lemma proof first statement derived proof theorem second statement proved differentiating fenchel identity duality show pair natural divergences defined functions moreover coincide clearly consider cost functions squared euclidean distance analogue definition gives classical bregman divergence covers bregman divergence framework best knowledge definitions depend crucially interplay transport divergence new use triple representation point definition consider function defined exponentially concave functions defined defined fenchel identity see vanish diagonal following generalization expression bregman divergence see theorem proposition expressions particular proof prove use fenchel identity starting proof similar show theorem namely proof using relation log log log next fenchel identity see pal wong using identities compute log log log computations convenient express solely terms either primal dual coordinates omit details computations lemma coordinate representations log log transport interpretation generalized pythagorean theorem using proposition give interesting transport interpretation expression generalized pythagorean theorem theorem let given let primal dual coordinates respectively proposition coupling monotone hence coupling optimal consider two suboptimal perturbations optimal coupling cyclical perturbation couple associated cost transposition couple keep coupling associated cost exponentially concave functions ask perturbation lower cost difference proposition nothing difference thus generalized pythagorean theorem gives information geometric characterization relative costs two perturbations examples consider portfolios example example portfolio let constantweighted portfolio cross entropy affine function also affine indeed log shannon entropy reason say portfolios transport map case given translation log log given primal coordinates see lemma log translation invariant property equivalent following invariance property lemma mapping fact difficult show property characterizes constantweighted portfolios among exponentially concave functions also see proposition chain rule analogous relative entropy pal wong example portfolio since log map scaling generalized diversityweighted portfolio example transport map composition scaling translation geometric structure induced section derive geometric structure induced given always impose regularity conditions assumption using primal dual coordinate systems definition compute explicitly riemannian metric primal connection confused euclidean gradient dual connection call induced geometric structure important fact information geometry connection necessarily right one use nevertheless duality always preliminaries differential geometric concepts riemannian metric affine connection refer reader chapters whose notations consistent computational convenience define geometric structure terms coordinate representations geometric structure determined independent choice coordinates intrinsic formulations refer reader chapter following definition makes sense general divergence manifold taken section definition induced geometric structure given coordinate system coefficients geometric structure given follows riemannian metric given gij assumption matrix gij strictly positive definite riemannian inner product length denoted respectively primal connection given exponentially concave functions iii dual connection given general divergence definitions first introduced define dual divergence dual connection primal connection primal dual connections dual respect riemannian metric see theorem divergence induces geometric structure may enjoy nice properties geometric structure induced bregman divergence shown curvatures primal dual connections vanish thus say induced geometry dually flat chapter show gives rise different geometry many interesting properties notations begin clarifying notations following convention see notation write depending coordinate system used primal dual coordinate representations computed lemma riemannian metric computed using primal dual coordinate systems explicit coordinate system use gij coefficients denote coefficients primal coordinates gij dual coordinates gij gij denoted inverses matrices gij gij respectively primal connection computed using primal coordinate system dual connection computed using dual coordinate system pal wong following notations useful define always adopt convention note involves portfolio second variable involves portfolio first variable partial derivatives given next lemma verified direct differentiation let kronecker delta lemma derivatives also note following easy fact used several times sincep partial derivatives zero lemma derviatives exponentially concave functions proof prove second formula proof first similar using write regard function recall note nth term sum omitted thanks formulas computations primal dual coordinates similar except change sign following often give details one coordinate system leave one reader last least let jacobian change coordinate map similarly let jacobian inverse map two jacobians inverses identity matrix denote transpose matrix riemannian metric intuition first compute riemannian inner product using euclidean coordinates let tangent space pal wong proposition let represented euclidean coordinates similarly hess proof proposition log differentiating two times setting gives first equality polarizing gives general case second equality follows chain rule theorem riemannian metric primal coordinate system riemannian metric given gij inverse given dual coordinate system riemannian metric given gij inverse given exponentially concave functions proof lemma lemma compute differentiating respect setting get gij lemma alternative expression gij expressing matrix form gij diag identity matrix invert use fact verified directly seen special case shermanmorrison formula using diag follows expanding matrix product pal wong proofs follow lines later use record following formulas remark lemma thus right hand side symmetric despite pearence similarly primal dual connections theorem primal dual connections primal coordinate system coefficients primal connection given gik gjk gik dual coordinate system coefficients dual connection given gik gjk gik exponentially concave functions proof prove leave reader notational convenience momentarily suppress computation later without comment differentiating one time evaluating simplifying get gij plugging finally plifying gik compute gim gjm gim remark interesting note although connections defined terms third order derivatives coefficients given terms portfolio normalized gradient pal wong curvatures well known see chapter induced geometric structure bregman divergence dually flat case geometry whenever simplex verify compute riemannchristoffel curvature tensors primal dual connections subsection adopt einstein summation notation see avoid writing lot summation signs curvature tensor connection defined smooth vector fields lie bracket given coordinate system coefficients given rijk satisfy rijk rijk theorem primal dual curvatures let curvature tensors primal dual connections respectively primal coordinates coefficients given rijk gik gjk dual coordinates coefficients given rijk gik gjk particular nonzero everywhere proof prove statements using suppressing argument follows next compute work exponentially concave functions combining rijk gik gjk see vanish suppose contrary values rijk fix letting gik gjk next let need dim get gik since arbitrary contradiction end section showing primal dual connections constant sectional curvature see definition definition ricci curvature corollary primal dual sectional curvatures primal dual connections constant sectional curvature respect particular primal dual ricci curvatures satisfy einstein condition ric proof primal connection constant sectional curvature respect zix ziy smooth vector fields see primal curvature tensor rijk implies sectional curvature claim ricci curvature follows immediately taking trace see example proof dual curvatures geodesics generalized pythagorean theorem armed primal dual connections formulate primal dual geodesic equations solutions primal dual geodesics studied section highlight section generalized pythagorean theorem theorem along way prove remarkable properties geometric structure pal wong primal dual geodesics note figure primal geodesic drawn straight line prove indeed case true dual geodesic dual euclidean coordinates let smooth curve denote time derivatives let primal dual coordinate representations say primal geodesic satisfies dual geodesic satisfies theorem primal geodesic equation primal coordinates dual geodesic equation dual coordinates theorem primal dual geodesics let primal geodesic trace euclidean straight line joining let dual geodesic let dual euclidean coordinate trace euclidean straight line joining prove leave reader proof makes use following lemmas lemma let let primal coordinates respectively consider differential equation exponentially concave functions defined log exists unique solution satisfying proof first note satisfies map also domain scaled also let maximal solution defined interval tmax since equation solved neighborhood tmax previous remark strictly increasing tmax hits tmax function solution desired properties fact claim lim sup defined inf tmax min min thus approaches least one fractions blows follows supt tmax suppose contrary let solution satisfying note exists fractions finite continuous near range contains open interval containing thus exists tmax allows extend range beyond contradicts maximality uniqueness theorem ode note smooth coefficients solution unique lemma let solution lemma consider curve given exponential coordinates primal geodesic moreover trace euclidean straight line joining pal wong proof primal geodesic verified directly differentiating plugging primal geodesic equation omit computational details see trace euclidean straight line consider euclidean representation solving gives expressing euclidean coordinates using get algebra hence exists together identity shows time change euclidean straight line proof theorem shown lemma pair points exists primal geodesic euclidean straight line remains observe geodesic unique indeed let primal geodesic solves primal geodesic equation consider initial velocity let varying well initial speed exists primal geodesic insthe form uniqueness theorem ode using dual coordinate system dual euclidean coordinate system proved similar way considering curve defined log exponentially concave functions connection projectively flat coordinate system geodesics straight lines time reparameterizations say geometric structure dually projectively flat projectively flat view theorem following corollary corollary manifold equipped geometric structure dually projectively flat flat gradient flows inverse exponential maps motivated recent paper relate primal dual geodesics gradient flows fix consider following gradient flows starting primal flow dual flow grad denotes riemannian gradient respect metric call primal flow dual flow verified easily since standard ode theory shown solutions defined lim lim words gradient flows converge unique minimizers theorem gradient flows primal flow time change primal geodesic dual flow time change dual geodesic pal wong recall concept exponential map consider primal geodesic starting initial velocity defined time define expq dual exponential map defined analogously corollary theorems following characterization primal dual inverse exponential maps corollary inverse exponential maps let exp exponential maps respect primal dual connections respectively prove theorem begin computing riemannian gradients computation somewhat tricky given appendix recall notations lemma riemannian gradients let primal coordinate system grad dual coordinate system grad proof theorem prove leave reader let primal representation primal flow starting lemma time exponentially concave functions constant proportionality depends independent follows comparing see primal flow time change primal geodesic generalized pythagorean theorem characterized primal dual geodesics ready prove generalized pythagorean theorem proof makes use riemannian gradients given lemma reason gradients appear correct scaling easier handle seen proof see proof theorem given consider primal geodesic dual geodesic let theorem proportional initial velocities two geodesics thus suffices prove sign claim established following two lemmas lemma sign proof lemma sign rearranging since scaling change sign may consider instead quantity get expanding pal wong lemma consider tangent vectors defined proof computation use primal coordinate system using definition riemannian inner product compute gij claim exponentially concave functions see write last expression simplified using identities gives claim finally expanding simplifying obtain desired identity two lemmas proof complete application mathematical finance subsection explain new information geometry applied finance financial background details refer reader consider sequential investment stock market stocks time let distribution capital market explicitly let market capitalization stock market weight stock let portfolio map defines investment strategy current state market distribute capital among stocks according vector example invest stock stock rest stock period perform necessary trading proportions capitals invested rebalanced one chooses identity map resulting portfolio called market portfolio denote market portfolio plays role benchmark let portfolio map generated exponentially concave function definition consider value resulting portfolio pal wong ots equa ghted severa ues boundary ves port geodes ang sense theorem beginning time also consider value market portfolio starting consider ratio value portfolio time value portfolio time called relative value portfolio suitable conditions see shown proposition fernholz decomposition log time sense cumulative measures volatility harvested portfolio see optimization quantity discussion assumes portfolio rebalances every period say every week practice due transaction costs considerations may want rebalance frequencies let exponentially concave functions three time points consider two ways implementing portfolio rebalance times rebalance time fernholz decomposition relative values two implementations time log log letting difference two values log log generalized pythagorean theorem sign difference determined angle dual geodesic primal geodesic figure illustrate result equalweighted portfolio stocks figure rebalancing time creates extra profit lies outside region shows convincingly rebalancing frequently always better even absence transaction costs importantly framework provides geometric way saying sequence volatile subsequence displacement interpolation final section consider displacement interpolation optimal transport problem formulated section refer reader chapter chapter introductions displacement interpolation time dependent transport problem let borel probability measures consider transport problem cost given suppose transport problem solved terms exponentially concave function letting monge optimal transport map given particular pushforward idea displacement interpolation introduce additional time structure want define action curves cost function given min minimum taken smooth curves satisfying pair minimizing curve gives pal wong map transporting along continuous path let defined minimizing curve pair want define way optimal transport map probability measures classical euclidean case cost action optimal transport map form ordinary concave function displacement interpolations linear interpolations see theorem theorem particular individual trajectories minimizing curves euclidean straight lines regarded geodesics flat geometry section formulate prove analogous statement transport problem lagrangian action portfolio interpolation begin defining appropriate action let smooth curve define exponential coordinate equivalently intuitively think time sense interpolation note define lagrangian action log take log argument alternative representation action log lemma min exponentially concave functions action minimized curve log tqi tqn particular minimizing curve proof fix smooth curve since log convex jensen inequality log log log log curve defined constant equality holds finally follows direct calculation displacement interpolation work following setting let borel probability measures let exponentially concave function satisfying assumption optimal transport map function let portfolio map generated consider flow defined minimizing curves log portfolio map defined following main result section interesting note displacement interpolation interpreted naturally linear interpolation portfolio terminal portfolio theorem displacement interpolation section consider setting pal wong portfolio map generated exponentially concave function defined log let let distributed according distributed according moreover optimal transport map transport problem iii endow geometric structure induced assume surjective fixed consider curve dual coordinates trace curve dual geodesic joining proof follows directly example clear coupling proposition graph map monotone proves optimal transport map iii write form see time change dual geodesic surjectivity assumption guarantees curve lies within range dual coordinate system another interpolation financial perspective another natural interpolation namely linear interpolation market portfolio example portfolio exponentially concave functions corresponding log generating function transport perspective market portfolio corresponds trivial transport map recall proposition iii portfolio exponential coordinate given argument theorem following result proposition consider geometric structure induced assume range dual coordinate system consider flow given interpolation dual coordinates trace curve time change dual geodesic appendix technical proofs proof theorem proof treat independent variables prove together begin observing see write switching coordinates using chain rule bit computation see equals consider given inf differentiating using see attains infimum rearranging log pal wong proves equality holds satisfies relation particular given next prove minimizer exists unique consider instead maximization quantity expanding switching euclidean coordinates equals quotient strictly concave function affine function right hand side strictly superlevel sets strictly convex see example shows minimizer unique exists let exists unique equality holds particular completes proof next prove smooth diffeomorphism since smooth injective inverse function theorem suffices show jacobian invertible invertibility jacobian follows fact gij strictly positive definite finally fenchel identity express form since cost function smooth see smooth well proof lemma prove first formula exponentially concave functions pute using grad second formula first prove claim see use compute follows pal wong compute using theorem symmetry grad second last equality used proof similar acknowledgements authors would like thank robin graham helpful discussions also thank anonymous referee suggestions improved paper references acciaio penkner schachermayer version fundamental theorem asset pricing superreplication theorem mathematical finance amari information geometry applications springer amari nagaoka methods information geometry american mathematical soc ambrosio gigli users guide optimal transport modelling optimisation flows networks springer amari novel approach canonical divergences within information geometry entropy banner fernholz relative arbitrage volatilitystabilized markets annals finance exponentially concave functions basseville divergence measures statistical data processing annotated bibliography signal processing penkner modelindependent bounds option pricesa mass transport approach finance stochastics booth fama diversification returns asset contributions financial analysts journal bouchey nemtchinov paulsen stein volatility harvesting diversifying rebalancing create portfolio growth journal wealth management boyd vandenberghe convex optimization cambridge university press bregman relaxation method finding common point convex sets application solution problems convex programming ussr computational mathematics mathematical physics calin geometric modeling probability statistics springer chambers zdanowicz limitations diversification return journal portfolio management klartag moment measures journal functional analysis mccann riemmanian interpolation inequality borell brascamp lieb invent math dillen nomizu vranken conjugate connections radon theorem affine differential geometry monatshefte mathematik dolinsky soner martingale optimal transport robust hedging continuous time probability theory related fields eguchi second order efficiency minimum contrast estimators curved exponential family annals statistics eguchi geometry minimum contrast hiroshima math erb harvey strategic tactical value commodity futures financial analysts journal erbar kuwada sturm equivalence entropic condition bochner inequality metric measure spaces inventiones mathematicae fernholz portfolio generating functions quantitative analysis financial markets avellaneda world scientific fernholz equity portfolios generated functions ranked market weights finance stochastics fernholz stochastic portfolio theory applications mathematics springer fernholz karatzas kardaras diversity relative arbitrage equity markets finance stochastics fernholz shay stochastic portfolio theory stock market equilibrium journal finance hallerbach disentangling rebalancing return journal asset management hazan agarwal kale logarithmic regret algorithms pal wong online convex optimization machine learning juditsky rigollet tsybakov learning mirror averaging annals statistics karatzas ruf trading strategies generated lyapunov functions arxiv kass vos geometrical foundations asymptotic inference john wiley sons kurose dual connections affine geometry mathematische zeitschrift lott villani ricci curvature spaces via optimal transport annals mathematics mahdavi zhang jin lower upper bounds generalization stochastic exponentially concave optimization proceedings conference learning theory matsuzoe geometry contrast functions conformal geometry hiroshima mathematical journal mccann convexity theory interacting gases equilibrium crystals phd thesis princeton university mccann convexity principle interacting gases advances mathematics murray rice differential geometry statistics crc press nagaoka amari differential geometry smooth families probability distributions univ tokyo tokyo japan metr oliker embedding given integral gauss curvature optimal mass transport adv math pal exponentially concave functions high dimensional stochastic portfolio theory arxiv pal embedding optimal transports statistical manifolds appear indian journal pure applied mathematics pal wong energy entropy arbitrage arxiv pal wong geometry relative arbitrage mathematics financial economics rockafellar convex analysis princeton landmarks mathematics princeton university press shima geometry hessian structures world scientific sternberg curvature mathematics physics dover strong generalizations functionally generated portfolios applications statistical arbitrage siam journal financial mathematics villani topics optimal transportation graduate studies mathematics american mathematical soc villani optimal transport old new springer willenbrock diversification return portfolio rebalancing commodity return puzzle financial analysts journal wong optimization relative arbitrage annals finance wong universal portfolios stochastic portfolio theory arxiv exponentially concave functions university washington padelford hall seattle washington usa soumikpal university southern california kaprielian hall los angeles california usa tkleonardwong
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onvergence tability gan dec naveen kodali jacob abernethy james hays zsolt kira college computing georgia institute technology atlanta usa prof hays zkira bstract propose studying gan training dynamics regret minimization contrast popular view consistent minimization divergence real generated distributions analyze convergence gan training new point view understand mode collapse happens hypothesize existence undesirable local equilibria game responsible mode collapse observe local equilibria often exhibit sharp gradients discriminator function around real data points demonstrate degenerate local equilibria avoided gradient penalty scheme called dragan show dragan enables faster training achieves improved stability fewer mode collapses leads generator networks better modeling performance across variety architectures objective functions ntroduction generative modeling involves taking set samples drawn unknown data generating distribution preal finding estimate pmodel closely resembles generative adversarial networks gan goodfellow powerful framework used fitting implicit generative models basic setup consists two networks generator discriminator playing repeated game setting goal reach equilibrium preal pmodel close alternating gradient updates procedure agd used achieve however process highly unstable often results mode collapse goodfellow calls deeper investigation training dynamics gans paper propose studying gan training dynamics repeated game players using algorithms lugosi discuss agd falls paradigm contrast much theory goodfellow arjovsky bottou recent developments nowozin arjovsky gulrajani based unrealistic assumption discriminator playing optimally function space step result consistent minimization divergence real generated distributions corresponds least one player using algorithm function space resulting game dynamics completely different cases nisan thus clear disconnect theoretical arguments used motivation recent literature actually happens practice would like point latter view still useful reasoning asymptotic equilibrium situation argue regret minimization appropriate way think gan training dynamics analyze convergence gan training new point view understand mode collapse happens start short analysis artificial case gan game section setting unique solution guaranteed convergence averaged iterates using algorithms shown standard arguments game theory literature make explicit critical previously widely known connection agd used gan training regret minimization immediately code https analysis applies simultaneous gradient updates procedure well yields novel proof asymptotic convergence gan training limit prior work result goodfellow required strong assumption discriminator optimal step however convergence results hold game objective function practical case deep neural networks used games global regret minimization equilibrium computation computationally hard general recent gametheoretic literature indicates agd end cycling mertikopoulos converging potentially bad local equilibrium conditions hazan hypothesize reasons cycling mode collapse observed gan training respectively section work explore cycling issue focus attention mode collapse problem contrast hypothesis prevalent view mode collapse instability arjovsky bottou results attempting minimize strong divergence training however argued earlier gan training agd consistently minimize divergence therefore theory suitable discuss convergence address stability issue next mode collapse indeed result undesirable local equilibrium natural question avoid make simple observation gan game mode collapse situations often accompanied sharp gradients discriminator function around real data points section therefore simple strategy mitigate mode collapse regularize discriminator constrain gradients ambient data space demonstrate improves stability using toy experiment one hidden layer neural networks gives rise new explanation wgan gradient penalties might improving stability gan training mitigating mode collapse problem keeping gradients discriminator function small data space motivation propose training algorithm involving novel gradient penalty scheme called dragan deep regret analytic generative adversarial networks enables faster training achieves improved stability modeling performance gulrajani stable training procedure across variety architectures objective functions provide short literature review several recent works focus stabilizing training gans solutions radford salimans require usage specific architectures modeling objectives che zhao significantly deviate original gan framework promising works direction metz arjovsky gulrajani impose significant computational overhead thus fast versatile method consistent stable training gans still missing literature work aimed addressing summarize contributions follows propose new way reasoning gan training dynamics viewing agd regret minimization provide novel proof asymptotic convergence gan training nonparametric limit require discriminator optimal step discuss agd converge potentially bad local equilibrium games hypothesize responsible mode collapse gan training characterize mode collapse situations sharp gradients discriminator function around real data points novel gradient penalty scheme called dragan introduced based observation demonstrate mitigates mode collapse issue heoretical nalysis gan training dynamics start brief description gan framework section discuss guaranteed convergence artificial case using algorithms make critical connection gan training process agd regret minimization section immediately yields novel proof asymptotic convergence gan training nonparametric limit consider practical case discuss agd converge potentially bad local equilibrium section characterize mode collapse situations sharp gradients discriminator function around real samples provides effective strategy avoid naturally leads introduction gradient penalty scheme dragan section end discussion comparison gradient penalties literature section background gan framework viewed repeated game consisting two players generator produces synthetic data given noise source discriminator trained distinguish generator samples real data generator model parameterized takes noise vector input produces synthetic sample discriminator model parameterized takes sample input computes interpreted probability real models selected arbitrary class functions practice gans typical rely deep networks cost functions defined log log complete game specified min max log log generator distribution pmodel asymptotically converges real distribution preal updates made function space discriminator optimal step goodfellow onvex concave case regret algorithms according sion theorem sion compact convex sets function convex first argument concave second min max max min equilibrium guaranteed exist setting players payoffs correspond unique value game neumann natural question find equilibrium simple procedure players use algorithms brd round players play optimal strategy given opponent current strategy despite simplicity brd often computationally intractable lead convergence even simple games contrast technique efficient provably works regret minimization players update parameters using algorithms easy show averaged iterates converge equilibrium pair nisan let first define algorithms definition algorithm given sequence convex loss functions algorithm selects sequence may depend previously observed said regret define apply learning problem equilibrium finding gan game follows generator imagines function loss function round similarly discriminator imagines loss function rounds play player computes average iterates equilibrium value game players suffer regret respectively one show using standard arguments freund schapire words almost optimal solutions game almost approximation factor given average regret terms condition former vanish hence guarantee convergence limit next define popular family algorithms definition follow regularized leader ftrl hazan selects round solving arg convex regularization function learning rate remark roughly speaking select regularization ftrl becomes online gradient descent ogd zinkevich ignoring case constraint violations ogd written simple iterative form typical gan training procedure using alternating gradient updates simultaneous gradient updates almost players applying online gradient descent notice objective function gans involves stochastic component two randomized inputs given round sampled data distribution standard multivariate normal respectively let write log log taking expectations respect define full game online training procedure still valid stochastic inputs equilibrium computation would proceed similarly round sample follow updates jxt jxt side note benefit stochastic perspective get generalization bound mean parameters rounds optimization celebrated conversion implies optimal value plus estimation error bounded expectation taken respect sequence samples observed along way randomness algorithm analogously applies well limitation result however requires fresh sample used every round summarize discussed subsection artificial case easy solve regret minimization standard result game theory online learning literature widely known gan literature instance salimans goodfellow discuss toy game show cycling behavior simple solution case average iterates made explicit critical connection regret minimization alternating gradient updates procedure used gan training goodfellow argue enough capacity limit updates made function space gan game considered thus analysis based regret minimization immediately yields novel proof asymptotic convergence gans require discriminator optimal step moreover connection regret minimization gan training process gives novel way reason dynamics contrast popular view gan training consistently minimizing divergence arises discriminator uses brd function space thus little actual training process gans result calls question motivation behind many recent developments like wgan gradient penalties among others improve training stability gans next subsection discuss practical case training instability arises provides necessary ideas investigate mode collapse new perspective convex case ocal equilibria practice choose deep neural networks function need convexconcave anymore nice properties case like existence unique solution guaranteed convergence regret minimization longer hold fact regret minimization equilibrium computation computationally hard general settings however analogous case optimization also intractable focus finding local minima look tractable solution concepts games recent work hazan introduces notion local regret shows players use smoothed variant ogd minimize quantity game converges form local equilibrium mild assumptions usual training procedure gans agd corresponds using window size formulation thus gan training eventually converge approximately local equilibrium described updates cycle leave future works explore equally important cycling issue focus former case definition local equilibrium pair called local equilibrium holds local equilibrium players much incentive switch strategy within small neighborhood current strategies turn attention mode collapse issue poses significant challenge gan training process training said resulted mode collapse generator ends mapping multiple vectors output assigned high probability real discriminator goodfellow hypothesize result game converging bad local equilibria prevalent view mode collapse instability gan training arjovsky bottou caused due supports real model distributions disjoint lying manifolds argument would result strong distance measures like getting maxed generator get useful gradients learn fact motivation introduction wgan arjovsky argued earlier gan training consistently minimize divergence would require using intractable algorithms hence theory suitable discuss convergence address instability gan training new view gan training process regret minimization closer used practice provides alternate explanation mode collapse existence undesirable local equilibria natural question avoid ode collapse radient enalties problem dealing multiple equilibria games avoid undesirable ones important question algorithmic game theory nisan work constrain gan game aim characterize undesirable local equilibria mode collapse effort avoid direction empirically studying multiple mode collapse cases found often accompanied discriminator function sharp gradients around real data points see figure intuitively makes sense definition mode collapse discussed earlier sharp gradients encourage generator map multiple vectors single output lead game towards degenerate equilibrium simple strategy mitigate failure case would regularize discriminator using following penalty strategy indeed improves stability gan training show results toy experiment one hidden layer neural networks figure figure demonstrate partly explains success wgan gradient penalties recent literature gulrajani improve training stability gans despite motivated reasoning based unrealistic assumptions however noticed scheme current form brittle discriminator end assigning real point noise probability real thus better choice penalty max times stochasticity seems help getting basin attraction bad equilibrium figure one hidden layer networks mnist left plot inception score time vanilla gan training right plot squared norm discriminator gradients around real data points experiment notice quantity changes mode collapse events figure one hidden layer networks mnist left losses players shown vanilla gan training right added regularization term penalize gradients around real data points notice improved stability finally due practical optimization considerations also observed gulrajani instead use penalty shown experiments still works long small perturbations real data likely lie true case image domain settings cases want discriminator assign different probabilities real training data noisy samples caution practitioners keep important point mind making choice penalty schemes effect constraining norm discriminator gradients around real points small therefore mitigate mode collapse situation refer gan training using penalty schemes heuristics dragan algorithm additional details use vanilla gan objective experiments penalty improves stability using objective functions well demonstrated section penalty scheme used experiments one shown equation use small noise possible find better ways imposing penalty however exploration beyond scope paper optimal configuration hyperparameters dragan depends architecture dataset data domain set experiments oupled ocal enalties several recent works also proposed regularization schemes constrain discriminator gradients ambient data space improve stability gan training despite figure one hidden layer networks mnist left inception score plot shown vanilla gan training right added regularization term penalize gradients around real data points notice mode collapse mitigated different motivations closely related approaches first show two approaches similar widely known literature introduced idea maintaining margin losses assigned real fake samples also impose lipschitz constraint two conditions together result situation following holds real fake sample pair roughly authors argue resulting discriminator function would gradients almost everywhere real fake samples section next gulrajani proposed extension address various shortcomings original wgan impose following condition point line real fake sample chosen independently random leads gradients almost everywhere real fake samples notice behavior similar discriminator function thus slight variation original algorithm refer methods coupled penalties side note also want point penalty actually follow claimed paper lemma gulrajani optimal discriminator gradients almost everywhere pairs sampled optimal coupling joint distribution therefore basis penalty equation arbitrary pairs real fake samples used fact adds credence theory regarding gradient penalties might mitigating mode collapse important distinction coupled penalties methods impose gradient constraints local regions around real samples refer penalty schemes local penalties coupled penalties impose gradient constraints real generated samples point potential issues arise adversarial training finding applications beyond fitting implicit generative models penalties depend generated samples prohibitive resulting class functions coupled penalties used highly restricted compared method affects modeling performance refer reader figure appendix section see effect algorithm works agd needs multiple inner iterations optimize generated samples anywhere data space change one iteration next contrast consistently regularize along real data manifold conclude appropriate constraining discriminator gradients mitigate mode collapse careful negative effects pointed issues coupled penalties local penalties help refer reader section experimental results figure swissroll experiment different phases training vanilla gan top middle dragan bottom real samples marked orange generated samples green level sets shown background yellow high purple low xperimental esults section compare modeling performance algorithm vanilla gan wgan variants standard setup section demonstrates dragan improved stability across variety architectures section show method also works objective functions appendix contains samples inspection missing plots additional results throughout use inception score salimans reliable metric literature sample quality measure performance nception cores using dcgan architecture dcgan family architectures designed perform well vanilla training procedure ubiquitous gan literature owing instability vanilla gan general settings use architecture model compare vanilla gan wgan wgangp wgans need discriminator iterations every generator iteration comparing modeling performance tricky address report two scores vanilla gan dragan one using number generator iterations wgans one using number discriminator iterations results shown figure samples included appendix figure notice dragan beats wgan variants configurations vanilla gan slightly better key point note algorithm fast compared wgans practice performance closer dragand case next section show move away specific architecture family vanilla gan training become highly unstable dragan penalty mitigates issue easuring stability performance across architectures ideally would want training procedure perform well stable fashion across variety architectures dcgans similar arjovsky gulrajani remove stabilizing components dcgan architecture demonstrate improved stability modeling performance compared vanilla gan training see appendix section however small set architectures clear improvement general address introduce metric termed bogonet score compare stability performance different gan training procedures basic idea choose random architectures players independently evaluate performance different algorithms resulting algorithm wgan dragang dragand vanilla gang vanilla gand score inception scores inception score plot figure comparison modeling performance table summary inception score statistics across architectures algorithm vanilla gan dragan final score mean std area curve mean std qual score total games good algorithm achieve stable performance without failing learn resulting mode collapse despite potentially imbalanced architectures experiment player assigned network diverse pool architectures belonging three different families mlp resnet dcgan demonstrate algorithm performs better compared vanilla gan training created instances hard games instance trained using algorithms similar conditions fixed number generator iterations gives slight advantage plot inception score changes time algorithm calculated average final inception scores area curve auc instances results shown table notice beat algorithms metrics indicates improvement stability modeling performance perform qualitative analysis verify bogonet score indeed captures improvements stability create another set hard architectures compare dragan vanilla gan training instance allotted points split bounty two algorithms depending performance perform well perform poorly get points nullify effect architectures however one algorithm achieves stable performance compared terms failure learn mode collapses assign higher portions bounty results judged two authors blind manner curves shown choice algorithm side randomized unlabeled vanilla gan received average score algorithm achieved average score correlates bogonet score earlier see appendix section additional details regarding experiment tability using different objective functions algorithm improves stability across variety objective functions demonstrate using following experiment nowozin show interpret gan training minimizing various appropriate game objective function used show experiments using objective functions developed forward reverse pearson squared hellinger total variation divergence minimization use hard architecture previous subsection demonstrate improvements stability algorithm stable cases except total variation case vanilla algorithm failed cases see figure two examples figure appendix five thus practitioners choose game objective larger set functions use dragan unlike wgans requires specific objective function reverse pearson figure inception score plots two divergence measures demonstrating superior stability algorithm onclusions paper propose study gan training process regret minimization contrast popular view consistent minimization divergence real generated distributions analyze convergence gan training new point view hypothesize mode collapse occurs due existence undesirable local equilibria simple observation made mode collapse situation often exhibits sharp gradients discriminator function around real data points characterization partly explains workings previously proposed wgan gradient penalties motivates novel penalty scheme show evidence improved stability using dragan resulting improvements modeling performance across variety settings leave future works explore ideas depth come improved training algorithms eferences martin arjovsky bottou towards principled methods training generative adversarial networks arxiv preprint martin arjovsky soumith chintala bottou wasserstein gan arxiv preprint nicolo lugosi prediction learning games cambridge university press nicolo alex conconi claudio gentile generalization ability learning algorithms ieee transactions information theory tong che yanran athul paul jacob yoshua bengio wenjie mode regularized generative adversarial networks arxiv preprint yoav freund robert schapire adaptive game playing using multiplicative weights games economic behavior ian goodfellow jean mehdi mirza bing david sherjil ozair aaron courville yoshua bengio generative adversarial nets ghahramani welling cortes lawrence weinberger eds advances neural information processing systems curran associates url http ian goodfellow nips tutorial generative adversarial networks corr url http ishaan gulrajani faruk ahmed martin arjovsky vincent dumoulin aaron courville improved training wasserstein gans arxiv preprint elad hazan karan singh cyril zhang efficient regret minimization games arxiv preprint elad hazan introduction online convex optimization foundations optimization panayotis mertikopoulos christos papadimitriou georgios piliouras cycles adversarial regularized learning arxiv preprint luke metz ben poole david pfau jascha unrolled generative adversarial networks corr url http von neumann zur theorie der gesellschaftsspiele mathematische annalen url http noam nisan tim roughgarden eva tardos vijay vazirani algorithmic game theory volume cambridge university press cambridge sebastian nowozin botond cseke ryota tomioka training generative neural samplers using variational divergence minimization advances neural information processing systems generative adversarial networks lipschitz densities url http corr alec radford luke metz soumith chintala unsupervised representation learning deep convolutional generative adversarial networks november url http arxiv tim salimans ian goodfellow wojciech zaremba vicki cheung alec radford chen improved techniques training gans corr url http maurice sion general minimax theorems pacific math junbo zhao michael mathieu yann lecun generative adversarial network arxiv preprint martin zinkevich online convex programming generalized infinitesimal gradient ascent proceedings international conference machine learning ppendix amples atent pace walks section provide samples additional experiment run celeba dataset figure samples experiment section shown figure radford suggest walking manifold learned generator expose signs memorization use dcgan architecture model mnist celeba datasets using dragan penalty latent space walks learned models shown figure figure results demonstrate generator indeed learning smooth transitions different images algorithm used figure modeling celeba dragan using dcgan architecture vanilla gan dragan wgan figure modeling using dcgan architecture figure latent space walk model learned mnist using dragan figure latent space walk model learned celeba using dragan dditional xperiments hidden layer network model mnist design simple experiment fully connected networks one hidden layer vanilla gan performs poorly even simple case observe severe mode collapses contrast algorithm stable throughout obtains decent quality samples despite constrained setup inception score inception score generator iterations generator iterations vanilla gan dragan figure one hidden layer network model mnist inception score plots vanilla gan dragan figure one hidden layer network model mnist samples aussians xperiment analyze performance dragan dataset seen figure approximately converge real distribution notice case seems overly constrained data space contrast dragan discriminator flexible improved wgan dragan figure comparing performance dragan dataset orange real samples green generated samples level sets shown background yellow high purple low tability across dcgan rchitecture variations dcgan architecture designed following specific guidelines make stable radford restate suggested rules use networks learn spatial downsampling discriminator upsampling generator remove fully connected hidden layers deeper architectures use batch normalization generator discriminator use relu activation generator layers except output layer uses tanh use leakyrelu activation discriminator layers show constraints relaxed using algorithm still maintain training stability present series experiments remove different stabilizing components dcgan architecture analyze performance algorithm specifically choose following four architectures difficult train case start base dcgan architecture apply changes constant number filters generator relu mlp generator tanh nonlinearities everywhere tanh nonlinearity generator leakyrelu mlp discriminator notice case algorithm stable vanilla gan training fails similar approach used demonstrate stability training procedures arjovsky gulrajani tanh activation generator constant filter discriminator figure comparing performance dragan vanilla gan training hard variations dcgan architecture tability across objective functions due space limitations showed plots two cases section show results five cases reverse pearson forward total variation squared hellinger figure comparing performance dragan vanilla gan training using different objective functions ogo etails used three families architectures probabilities dcgan resnet mlp next parameterized family create additional variation instance dcgan family result networks without batch normalization leakyrelu tanh nonlinearities number width filters latent space dimensionality possible variations experiment similarly number layers hidden units layer mlp chosen randomly resnets chose depth randomly creates set hard games test stability given training algorithm showed qualitative analysis inception score plots section verify bogonet score indeed captures improvements stability show examples bounty splits done plots figure scored averages shown dragan vanilla gan order
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uplink channel estimation data transmission cran lens antenna arrays feb reuben george stephen student member ieee rui zhang fellow ieee mmwave communication network densification hold great promise achieving highrate communication wireless networks cloud radio access network cran remote radio heads rrhs coordinated central unit deployed serve users distributed manner costeffective solution achieve network densification however operating large bandwidth mmwave frequencies digital fronthaul links cran would easily saturated large amount sampled quantized signals transferred rrhs tackle challenge propose paper new architecture mmwavebased cran advanced lens antenna arrays rrhs due energy focusing property lens antenna arrays effective exploiting angular sparsity mmwave channels thus help substantially reducing fronthaul rate simplifying signal processing rrhs even channels consider uplink transmission mmwave cran lens antenna arrays propose quantization bit allocation scheme multiple antennas rrh meet given fronthaul rate constraint propose channel estimation technique exploits energy focusing property lens array implemented low complexity finally compare proposed mmwave cran using lens antenna arrays conventional cran using uniform planar arrays rrhs show proposed design achieves significant throughput gains yet much lower complexity index radio access network communication lens antenna array channel estimation fronthaul constraint antenna selection quantization bit allocation ntroduction etwork densification increasing densities base stations bss points aps deployed mmwave communication exploiting large unused bandwidth higher frequencies radio spectrum two key strategies achieving orders magnitude data rate improvement required future wireless communication networks one hand cloud radio access paper presented part ieee global communications conference globecom singapore stephen national university singapore graduate school integrative sciences engineering national university singapore singapore also department electrical computer engineering national university singapore singapore reubenstephen zhang department electrical computer engineering national university singapore singapore elezhang network cran distributed remote radio heads rrhs deployed close users coordinated central unit joint processing provides way achieving network densification thanks centralized resource allocation joint signal processing rrhs cran achieves significant improvements spectral efficiency energy efficiency compared conventional cellular network rrhs simple relay nodes transmit receive baseband signals fronthaul links hand advances radio frequency circuits wireless communication mmwave bands emerged promising technology achieve communications due large bandwidth available beamforming gain brought possibility deploying large number antennas transceivers thanks small wavelengths thus cran integrated mmwave communication achieves double goals network densification ample bandwidth time future wireless networks however dense cran operating large mmwave bandwidth digital fronthaul links would easily saturated large volume sampled baseband signals need transmitted rrhs thus crucial find solutions reduce transmission rate required fronthaul link crans operating conventional cellular frequency bands considerable body prior work investigated various techniques data compression rrhs achieve fronthaul rate reduction channel estimation networks coordinated bss also considered prior work random matrix theory used derive approximate lower bound uplink ergodic achievable rate system multiple users bss considered approach uplink cran proposed various schemes optimize ergodic achievable sum rate subject backhaul constraints techniques compression channel estimation applicable relatively small bandwidth compared mmwave typically involve complex signal processing cooperative signal compression across rrhs difficult implement mmwave systems due practical cost complexity considerations training sequence design cran considered problem minimizing wavelengths antenna element lens wavelengths wavelengths focusing antennas selected signal sampling quantization thus use lens antenna arrays potentially achieve significant reduction fronthaul rate requirement transmitting quantized signals interference among users joint decoding uplink transmission however due finite fronthaul rate constraint rrh crucial design antenna selection signal quantization schemes maximize achievable user rates since rrhs cran typically simple relay nodes paper consider simple uniform scalar quantization sampled baseband signals performed independently antennas selected rrh major contributions paper summarized follows fig illustration lens antenna array ing length maintaining local orthogonality among training sequences users considered however fronthaul constraints rrhs ignored frequencyselective mmwave channels channel estimation hybrid precoding considered approach represent channel taps angular domain corresponding set quantized angles use sparse signal processing techniques estimate relevant parameters paper propose new architecture mmwave cran leveraging use advanced lens antenna arrays rrhs lens antenna array consists electromagnetic lens energy focusing capability integrated antenna array whose elements located focal surface lens see fig amplitude response lens array expressed sinc function terms angles plane waves incident locations antenna elements hence appropriately designing locations antenna elements focal surface lens array capable focusing energy uniform plane wave arriving particular direction onto specific antenna element subset elements moreover due multipath sparsity mmwave channels lens array used achieve capacity mmwave channel multiple antennas via technique called path division multiplexing uses simple modulation even transmission wideband frequencyselective channels low signal processing complexity lens antenna arrays angular domain sparsity mmwave channels transformed spatial domain enables lower complexity signal processing channel estimation data transmission moreover cran since angles arrival signals different users typically independent thus different rrh user signals effectively separated small number introduce new architecture cran mmwave frequencies using lens antenna arrays rrhs taking consideration elevation azimuth angles arrival signals users uplink transmission propose simple based antenna selection rrh quantization bit allocation algorithm selected antennas minimize total quantization noise power based estimates received signal power different antennas subject fronthaul rate constraint perfect channel state information csi show proposed system lens antenna arrays achieve better performance compared conventional cran using uniform planar arrays upas rrhs orthogonal frequency division multiplexing ofdm transmission users quantization algorithm employed cases imperfect csi propose approximate mmse beamforming exploiting energyfocusing property lens antenna arrays proposed scheme user data streams estimated channel gains larger certain threshold chosen beamforming interference streams also approximated thresholding channel gain estimates proposed bit allocation rrhs channel estimation data transmission singlecarrier modulation compare proposed system conventional cran upas via simulations show fronthaul constrained proposed system achieve significant gains much lower signal processing complexity training overhead compared benchmark rest paper organized follows section present system model proposed mmwave cran lens arrays carry analysis achievable rates perfect imperfect csi proposed bit allocation channel estimation schemes section iii gives brief description benchmark cran system upas ofdm transmission section compare proposed system benchmark system via simulations finally section concludes paper notation paper denotes equality definition means distributed cardinality finite set denoted denotes elements sets real complex integer matrices denoted respectively denote set real numbers positive integers respectively denotes kronecker delta function normalized sinc function defined sinc sinc sin imaginary unit denoted scalars denoted letters vectors matrices denoted letters respectively denotes smallest integer greater equal denotes largest integer less equal denotes magnitude denotes phase radian vector kxk denotes euclidean norm vector elements equal denoted dimension implied context vectors matrices denotes transpose denotes conjugate transpose hermitian matrix denotes sum diagonal elements trace linearly independent columns denotes denotes kronecker product two matrices denotes identity matrix dimension diagonal matrix elements main diagonal denoted diag block diagonal matrix blkdiag denotes circularly symmetric complex gaussian cscg distribution centered covariance denotes uniform distribution interval ystem odel study uplink transmission dense cran cluster see fig rrhs denoted cluster sectorized rrh sectors covering sector served lens antenna array antenna elements denoted set denotes set sectors use without superscript denote set antenna elements sectors rrh rrh connected via individual fronthaul link finite capacity bits per second bps users cran cluster denoted users rrhs share bandwidth communication depending user location signals incident one sector rrh model leakage interference gaussian noise assume total number sectors cran greater equal number users users transmit mmwave channel bandwidth fronthaul link rrh central unit mmwave wireless link user lens array cran cluster fig schematic cran channel user rrh paths denoted channel coefficient vector time index user rrh expressed using geometric channel model dmax denote respectively complex gain delay symbol periods corresponding path array response elevation azimuth angles arrival path denoted respectively consider sector rrh equipped rectangular lens plane dimensions normalized along respectively lens followed antenna array elements placed focal surface lens hemisphere around lens center taken origin fig radius equal focal length lens see fig let antenna element indexed pair indexes denotes index elevation direction along focal surface lens denotes index azimuth direction along focal surface ray drawn center lens antenna element let denote azimuth angle made ray maximum azimuth angles negative positive respectively similarly let denote elevation angle made ray maximum elevation angles covered antenna array negative positive respectively antennas placed indexes run integers sin sin since user signals received one sector every rrh refer channel user rrh instead particular sector rrh convenience dimensions assumed rrhs convenience power probing guard guard antenna interval pilot symbols interval selection dmax data symbols power est baseband fig frame structure uplink transmission lens antenna arrays lens antenna array power est switches dmax yqm baseband via fronthaul yqm bit antenna selection thus elevation angles antenna elements related indexes sin fig architecture rrh lens antenna array sin sin next index index runs integers uniform scalar quantization bit allocation cos sin cos sin azimuth angles antenna elements related rrhs stage users transmit constant indexes sin cos amplitude signals duration dmax cos sin cos sin amplitude response lens array element rrh obtains estimate average received power uniform plane wave incident antenna either agc circuitry using elevation azimuth angles expressed analog power estimators consider rrh performs uniform real imaginary components complex baseband samples received antenna sinc sin denoted using sinc cos sin bits symbols antenna forwarded rrh antenna selected subsequent channel training data transmission following since rrhs fronthaul links design resulting quantized samples finite capacities must quantize received signals expressed forwarding rrhs typically simple nodes limited processing capability consider perform uniform independently antenna represents quantization error modeled bit allocation adapted channel coherence random variable mean zero variance given interval shown fig rrhs perform bit allocation based estimated received power levels antenna element obtained converting signals baseband either using feedback automatic quantization error assumed uncorrelated gain control agc circuitry means analog power performed independently estimators implemented using antenna sample filters envelope detectors consider nyquist transmission users frame duration rate sampling transmission rate required forward quantized signals antennas symbol periods denotes minimum coherence bps time among channels frame must exceed fronthaul capacity rrh uses estimate average received power divided following three stages shown fig computed received signals stage power probing stage duration users perform subsequent channel training data transmit constant amplitude signals order enable transmission stages consider design quantization rrhs perform bit allocation antenna selection bit allocation minimize total noise power channel training stage duration users antennas subject fronthaul capacity constraint transmit pilot symbols performs channel rrh captured following optimization problem estimation selected antennas using quantized signals forwarded rrhs minimize data transmission stage duration users transmit data quantized forwarded subject rrhs decoding stages separated guard intervals dmax symbols see fig dmax denotes maximum delay spread channels note performed using following describe stage detail converters adcs since integers problem however variables relaxed following relaxed problem min convex since objective function constraint linear thus following proposition proposition optimal solution problem given max table lgorithm quantization bit allocation initialize tolerance repeat compute according set else set end use compute initialize find bisection similar compute final integer solution problem according converged proof please refer appendix let denote set antennas allocation according optimal solution problem proceed construct feasible integer solution original problem rounding follows appropriate threshold notice decreased would rounded making difficult satisfy constraint vice versa hence suitable found bisection interval time evaluating constraint updating accordingly observe optimal solution allocates bits antenna higher estimated power desirable since antennas receive stronger signals users likely useful information decoded algorithm summarized table denote final set selected antennas rrh assume fronthaul capacities number selected antennas rrh least since always feasible recover users signals via linear processing independent channels let qtot denote total number selected antennas ease exposition refer selected antennas streams use index qtot denote stream stream corresponds antenna sector rrh relaxed problem assume objective function defined well path delay compensation achievable perfect csi subsection describe operations assuming perfect csi rrhs equipped lens antenna arrays propose users transmit simultaneously modulation data symbols transmitted user given transmit power complex data symbol user using quantized symbols received corresponding stream expressed dmi assume decodes users data symbols via linear processing path delay compensation received signals stream described let arg denote strongest path among arriving stream selected antenna user due response lens array delayed versions particular user signals different angels arrival focused different antenna elements rrh general thus ensure symbols user undergone strongest path gain stream combined synchronized manner stream quantized symbols corresponding antenna rrh advanced delay dmi corresponding path obtain delay compensated signal dmi user given dmi dmi dmi dmi order write summation paths terms delay differences dmi dmi delay maximum gain path dmi define antenna user pair new channel coefficient dmi dmi equivalent channel coefficient sum coefficients corresponding path lmi user antenna delay difference maximum gain path user antenna expressed interference isi interference iui dmi dmi defined denote awgn quantization noise samples shifted dmi symbol periods note depends user whose maximum gain path used reference second third terms represent user delayed symbols interfering symbols users respectively collecting signals streams written vector form vectors dimension qtot consider performs linear receive beamforming beamforming vector cqtot construct estimate user symbol treating interference gaussian noise signal ratio sinr decoding given top next page iqm cqtot blkdiag qtot tot diag fined since transmit powers users fixed maximized according minimum mean squared error mmse criterion denotes covariance noise interference terms case sinr becomes assuming cscg distribution quantization noise lower bound achievable sum rate users thus given rlens next subsection consider channel estimation extend achievable rate analysis case imperfect csi channel estimation achievable imperfect csi since know csi priori needs estimate csi using pilot signals sent users quantized forwarded rrhs channel users transmit known pilot estimation stage see fig symbols given let dmax dmax denote vector channel taps user antenna vector quantized symbols ctp received channel estimation stage expressed ctp dmax toeplitz matrix constructed consecutive shifts pilot symbols users denotes awgn ctp itp denotes quantization noise also ctp dmax hti dmax sufficient estimate find channel coefficients corresponding tap delays assume prior knowledge probability distribution function pdf elements treated unknown constants estimate given arg min define since component vector dmax linear combination independent random variables second term modeled cscg random vector mean covariance due central limit xph theorem thus note dmax must satisfied solution exist mse estimate given xph minimized xph cik dmax constant example construction translates condition dmax satisfied user training sequence orthogonal every user training sequence ideal property ensured using sequences equal transmit power case dmax thus xph consequently estimate reduces mpitp dmax according tap estimated correlating received signal vector training sequence corresponding user shifted corresponding delay tap due energy focusing property lens antenna array different directions arrival users signals different rrhs stream typically one path corresponding particular user corresponding particular tap delay would dominate paths thus magnitudes channel coefficients vastly different depending angle arrival users signals directly using estimates taps according may ineffective overcome issue propose approximate linear mmse beamforming exploiting sinc response lens array user select streams contain least one dominant estimated tap perform approximate linear mmse beamforming selected streams thresholding channel estimates explained notice data decoding path delay compensation detailed section term corresponds channel coefficient maximum path gain user antenna due energy focusing property lens angle arrival user signal typically streams means maximum gain path user would dominate paths particular stream provided antenna corresponding stream response peaks one angles arrival user paths since know actual channel gains aim find set streams user magnitude maximum estimated channel gain exceeds certain threshold first estimate tap delays maximum gain paths user every stream denoted computed estimate follows arg max dmax element vector estimate essentially choose tap delay corresponding estimate largest magnitude estimated delay maximum gain path stream stream estimate channel coefficient corresponding estimated maximum gain path given dmax element vector estimate data transmission stage selects set streams estimated channel gains corresponding estimated maximum gain paths larger given threshold define least one arg max suitable threshold also let jik least one stream selected decoding user signal thus data transmission stage reduced set streams user given set streams performs path delay compensation section estimated delays maximum gain paths instead true delays dmi delay compensated signal expressed terms delay differences taps similar isi iui defined beamforming observations also placed symbol delay differences indicate estimated tap delay channel coefficients defined similar given dmi note denote true channel coefficients correspond delay difference estimated delay vector form written following subsection briefly describe benchmark scheme conventional cran rrhs use upas users transmit using ofdm vectors dimensions linear mmse beamformer cik user path delay compensation estimated tap delay maximum gain path stream given resulting sinr given top next page thus proposed channel estimation path delay compensation approximate linear mmse receive beamforming user lower bound effective users cran given covariance matrix interference terms diag however compute beamformer since estimates computed selected streams avoid using noisy estimates approximates isi iui terms thresholding estimates similar thresholding estimates denote given otherwise dmax element vector estimate note due response lens antenna array stream thresholded estimates usersh delay idifferences let jik cik denote vector thresholded channel estimates applies approximate linear mmse iii enchmark onventional cran ofdm upa case sector rrh assumed equipped rectangular upa physical dimensions adjacent antenna elements array separated distance equal half wavelength let denote set antennas sector rrh case denote set antenna elements number antenna elements per sector given denotes number antennas along upa number antennas along upa array response vector written exp sin exp sin ray along response linear exp cos sin exp cos sin response linear array along total channel bandwidth divided orthogonal scs equal width denoted uplink transmission protocol similar fig except users transmit symbols length symbol blocks using ofdm dmax length cyclic prefix hence guard intervals shown fig implicit case due discarded assume users transmit scs fully exploiting spatial multiplexing assumed knows quantization noise variances stream transmitted rrhs power probing stage every frame period along information number bits antenna since quantities change within frame general multiple ofdm symbols power probing channel estimation data transmission stages convenience assumed integer multiples ofdm symbols power probing channel estimation data transmission stages frame respectively let denote tth ofdm symbol transmitted user corresponding signals given inverse discrete fourier transform idft denotes dft matrix columns similar exp exp section rrhs perform antenna selection bit allocation received signals forward let denote set selected antennas rrh case notice upas selected set antennas quantization bit allocation rrh different lens antenna array due lack lens focusing let similar section index selected antennas streams stream index corresponds antenna rrh dft awgn dft quantization noise since element linear combination elements due central limit also theorem max defined similar stacking similar section estimate given note estimate exists dmax mse estimate minimized dmax constant due construction translates requirement dmax channel estimation channel estimation stage let denote pilot symbol transmitted user tth ofdm symbol also let diag diagonal matrix constructed pilot symbols scs vector channel coefficients user antenna dmax channel coefficient vector similar section dmax denotes first dmax columns received signal vector corresponding ofdm symbol antenna removing applying dft expressed let unitary matrix diag diagonal matrix unit amplitude training symbols satisfying ensured choosing diag block theorem user index written holds dmax estimate written dmax note gives channel estimates frequency domain estimates user given applying transformation length dmax starting index dmax ending dmax since cscg also maximum likelihood estimate achievable let denote data symbol transmitted user received signal corresponding stream expressed components dfts awgn quantization noise similar collecting received signals streams meters rrh user meters meters vectors dimensions perfect csi linear mmse beamforming vector maximizes receive sinr user treating interference users noise given blkdiag tot diag tot tot sinr user thus given users upas ofdm transmission given hand estimated csi computes linear mmse receiver matrix components computed transforming estimates outlined earlier resulting sinr given thus effective users upas ofdm transmission given imulation esults simulations consider part general sectorized cran illustrated fig rrhs located corners hexagon side meters along users randomly located common fig example rrh user layout simulations region covered one sector rrh shown fig rrhs assumed height meters users height vary size lens rrhs upas rectangular length lens along length along height rrhs greater maximum height users minimum distance along ground user rrh assumed based setup maximum coverage angles lens array elevation direction chosen corresponding angles azimuth direction due sectorization leads antennas per sector rrh lens array upa respectively carrier frequency ghz mmwave channel shared users rrhs bandwidth mhz maximum delay spread channel translates dmax symbols path loss user rrh separated distance modeled number paths user rrh equal equal probability time delays paths generated exponential distribution mean truncated relative power levels paths given exp gaussian standard deviation representing perpath shadowing thus watt received power rrh user accounting path loss phase shifts path generated uniformly randomly interval elevation angles arrival path user rrh pendent generated los angle elevation user rrh similarly azimuth angles arrival independent generated table average number selected antennas per rrh fronthaul capacity per sector gbps lens lens perfect fronthaul perfect csi perfect fronthaul perfect csi lens constrained fronthaul perfect csi constrained fronthaul perfect csi fronthaul capacity per sector gbps fig average rate per user common fronthaul capacity per rrh sector perfect csi dbm los angle azimuth power spectral density awgn additional noise figure rrh interference assumed equal dbm sectors fronthaul capacity rrhs benchmark scheme using ofdm upas rrhs number scs length dmax minimum coherence time channels taken frame assumed symbols long corresponds ofdm symbols proposed system lens arrays choose dmax benchmark system upas ofdm order make fair comparison assume dmax cases results averaged random user locations channel realizations figs plot average throughput per user common fronthaul capacity per sector users transmit power dbm fig shows performance assuming perfect csi without fronthaul constraints rrhs without fronthaul constraints perfect csi observed performance benchmark system ofdm upas close proposed system lens difference attributed loss due insertion case ofdm transmission upas lens array redistributes incident energy among antenna elements depending angle arrival plane waves reasonable expect offer large gain achievable rate constraint fronthaul however since transmission users required isi iui mitigated without insertion processing use lens arrays greatly simplify signal processing required compared system average per user average per user lens perfect fronthaul estimated csi perfect fronthaul estimated csi lens constrained fronthaul estimated csi constrained fronthaul estimated csi common fronthaul capacity per sector gbps fig average rate per user common fronthaul capacity per rrh sector estimated csi dbm overall complexity performing linear mmse beamforming users scs fronthaul constraint order proposed system lens arrays complexity offering reduction factor fronthaul constrained case observed fig proposed system lens arrays provides significant throughput gains conventional due energy focusing property lens makes antenna selection quantization bit allocation much effective lens arrays compared upas antenna selection bit allocation scheme antennas larger received power selected allocated bits compared antennas lower received power table compares number antennas selected rrhs cases lens focuses incident energy onto antenna elements selection bit allocation algorithm tends select elements large received power thus although number antennas selected cases less similar fronthaul capacity low selection bit allocation much effective lens arrays compared upaofdm received energy distribution less similar across antenna elements selection bit allocation thus adversely affect achievable rate compared case fronthaul constraint fig shows performance proposed system estimated table iii average number streams selected per user reduced mmse estimated csi lens antenna array fronthaul capacity per sector gbps number streams csi thresholding channel estimates given choose observed proposed channel estimation scheme via thresholding channel estimates system lens antenna arrays offers significant gains benchmark system quantization antenna selection rrhs adversely affects channel estimation well data transmission benchmark system fronthaul constrained hand due energy focusing property lens stronger channels relatively less affected quantization since channels users rrhs sparse angular domain lens converts angular domain sparsity spatial domain sufficient use estimates stronger channels retain gains subsequent mmse beamforming upaofdm system thresholding effective since energy distributed uniformly antenna elements way distinguishing significant channels spatial domain table iii shows average number streams selected thresholding channel estimates proposed reduced approximate mmse beamforming lens arrays rrhs observed total number streams potentially even order magnitude less total number streams upaofdm case see table leads huge reduction complexity receive beamforming cran since proposed system complexity kimax imax maximum number streams user benchmark moreover gain throughput achieved much lower training overhead dmax proposed system compared least dmax benchmark practice usual onclusion paper introduce new architecture mmwave cran lens antenna arrays rrhs propose bit allocation scheme rrhs based estimated received power levels antennas show fronthaul constrained proposed system uses transmission users combined path delay compensation reducedsize mmse beamforming achieve large throughput gains conventional system uses upas rrhs along ofdm transmission users mmse combining propose simple yet effective channel estimation scheme proposed system exploits unique energy focusing property lens array show proposed system still offers considerable advantage benchmark system imperfect csi moreover gains achieved much lower signal processing complexity since sufficient process considerably fewer number streams proposed system compared benchmark shows proposed system lens antenna arrays promising candidate future evolution cran operating mmwave frequencies ppendix problem convex strictly feasible optimally solved solving dual problem let denote lagrange multiplier constraint problem lagrangian given dual function given minbm differentiating respect setting derivative equal zero since dual variable must lie interval moreover according increases decrease making constraint feasible vice versa thus optimal solves dual problem found simple bisection search optimal solution problem obtained given proof proposition thus completed eferences andrews ieee sel areas vol june zhou optimized backhaul compression uplink cloud radio access network ieee sel areas vol june liu zhang optimized uplink transmission spatial compression forward ieee 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training design based graph coloring ieee trans wireless vol gao dai wang channel estimation millimeterwave massive mimo hybrid precoding fading channels ieee communications letters vol jun venugopal alkhateeb prelcic heath channel estimation hybrid wideband millimeter wave systems ieee sel areas vol zeng zhang chen electromagnetic antenna enabled massive mimo performance improvement cost reduction ieee sel areas vol june zeng zhang millimeter wave mimo lens antenna array new path division multiplexing paradigm ieee trans vol apr communications lens antenna array ieee wireless communications vol zeng yang zhang millimeter wave mimo lens antenna online available http kwon lim min chae massive mimo systems fabrication issues codebook design ieee trans microw theory vol july yang zeng zhang channel estimation millimeter wave mimo communications lens antenna arrays ieee trans veh appear narasimhamurthy tepedelenlioglu antenna selection systems channel estimation error ieee trans veh vol jun tse viswanath fundamentals wireless communication cambridge university press kay fundamentals statistical signal processing estimation theory upper saddle river usa popovic generalized polyphase sequences optimum correlation properties ieee trans inf theory vol jul sesia toufik baker lte umts long term evolution theory practice wiley publishing chi gomaa calderbank training signal design tradeoffs systems ieee trans wireless vol jul samimi rappaport statistical channel model non line sight urban channels proc ieee globecom
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submitted annals statistics arxiv nov consistency sparsity sliced inverse regression high dimensions qian zhigen jun harvard temple provide framework analyze phase transition phenomenon slice inverse regression sir supervised dimension reduction technique introduced mild conditions asymptotic ratio lim phase transition parameter sir estimator consistent dimension greater propose diagonal thresholding screening sir algorithm method provides estimate covariance matrix conditional expectation var desired dimension reduction space obtained multiplying inverse covariance matrix certain sparsity assumptions covariance matrix predictors loadings directions prove consistency estimating dimension reduction space high dimensional data analysis extensive numerical experiments demonstrate superior performances proposed method comparison competitors introduction continuous multivariate random variable subspace called effective dimension stands independence reduction edr space mild conditions cook intersection edr spaces edr space denoted called central space many algorithms proposed find subspace assumption dims line research commonly known sufficient dimension reduction sliced inverse regression sir first yet widely used method sufficient dimension reduction due simplicity computational efficiency generality asymptotic properties sir particular interest last two decades consistency sir proved fixed hsing carroll lin research supported center mathematical sciences applications harvard university zhao research supported nsf grant liu research supported nsf grant nih grant msc subject classifications primary secondary keywords phrases dimension reduction random matrix theory sliced inverse regression lin zhao liu zhu zhu fang later zhu obtained consistency similar restriction also appears two recent work see zhong jiang liu common strategy pursued many recent researchers make sparsity assumptions predictors play role explaining predicting apply various regularization methods instance nachtsheim applied lasso tibshirani dantzig selector candes tao elastic net zou hastie respectively solve generalized eigenvalue problems raised variety sdr algorithms however piece jigsaw missing understanding sir dimension diverges increases sir break similar question asked variety sdr estimates cook paper prove certain technical assumptions sir estimator consistent lim result inconsistency provides theoretical justifications imposing certain structural assumption sparsity high dimensional settings behavior sir high dimension called phase transition phenomenon similar principal component analysis pca unsupervised counterpart sir extension however means trivial samples sliced bins according order statistics sliced samples neither independent identically distributed difference increases difficulty significantly paper provide new framework study phase transition behaviour sir technical tools developed potentially extended study phase transition behaviour sdr estimators second part article aims extending original sir scenario dimension exp based equation section central space estimated column space col consistent estimate precision matrix estimate space col var estimate col column space var propose diagonal screening procedure based new univariate statistics varh diagonal elements var motivated recent work sparse pca johnstone ranking predictors according magnitude varh decreasingly choose set consisting first predictors active predictors sir procedure subsequently applied selected predictors estimate column space var col matrix formed top eigeni embed filling entries outside vectors chosen row set denote new matrix estimate central space defined col name algorithm diagonal thresholding sir prove dtsir consistent estimating central space certain regularity conditions extensive simulation studies show performs better competitors computationally efficient rest paper organized follows section briefly describe sir procedure introduce notations section brief review existing asymptotic results sir procedure state theorems discuss phase transition phenomenon sir section propose method show consistent high dimensional data analysis section provide simulation studies compare competitors concluding remarks discussions put section proofs presented appendices preliminaries notations sliced inverse regression consider multiple index model noise unknown link function without loss generality assume although matrix identifiable space spanned called column space denoted col might identified proposed sliced inverse regression sir procedure estimate central space col without knowing briefly summarized follows given samples sir first divides slices according order statistics data double subscript refers slice number refers order number sample slice concomitant let sample mean slice let mean samples var estimated ease notations arguments assume log log throughout article lin zhao liu based observation col col col sir estimates central space col throughout article matrix formed top eigenvectors assume fixed largest eigenvalue bounded away order sir result consistent estimate central space imposed following two conditions linearity condition linear combination coverage condition dimension space spanned central curve equals dimension central space notations let interval note intervals depend order statistics thus random product sample space define random variable density function matrix denotes formed restricting rows columns articular denotes formed restricting columns matrix embed putting entries outside denote new matrix similar notations apply vectors two positive numbers let max let min let hard thresholding function throughout article used denote generic absolute constants though actual value may vary case case vector denote entry let two vectors dimension angle two vectors denoted two sequences let stand positive let stand lim abnn consistency sir order control behavior sir need impose following boundedness condition predictors covariance matrix addition tail condition joint distribution also need condition central curve boundedness condition exist positive constants minimal maximal eigenvalues respectively central curve finite fourth moment stable defined respect definition two positive constants let collection partition satisfying central curve called stable respect exist positive constants central space partition var var central curve sliced stable stable positive constant remark note need hold unit vectors central space rescaling considering orthogonal decomposition general space respect central space complement easy see sliced stability implies holds true vector particular following two useful consequences choosing position var var coordinate central curve since equation holds unit vector var kvar lin zhao liu remark suppose samples respectively one let defined similarly hand classic consistent estimator ofpvar hand expect estimate var consistent must require average loss variance slice decrease zero increases definition simply choose decreasing rate power would easily seen smooth compactly supported holds automatically sense general curve random variable sliced stability condition smoothness central curve tail distribution surprised since work consistency sir estimate requires kind smoothness central curve tail distribution control popular smoothness tail condition might one proposed hsing carroll later used zhu zhu proof consistency sir explained let collection partitions first assumed central curve satisfies following smoothness condition lim sup second assumed exists function changing tail condition slightly stronger condition neykov proved modified condition implies sliced stability condition ready state main results theorem conditions proof theorem deferred appendix direct consequence theorem observe may choose log right hand side equation converges consistent estimate thus theorem implies remark convergence rate note convergence rate depends choice may seem desirable might different first glance since convergence rate col may expect convergence rate col depend choice fact proof theorem easily check first term convergence rate second term rate since share column space interested estimating convergence rate second term matter provided large enough integer may depend depend theorem hold categorical continuous response variable theorem conditions assuming lim probability converging one define distance two subspaces operator norm frobenius norm difference satisfy simple linear algebra shows col span let matrix formed top generalized eigenvectors recall eigenvalue assumed bounded away therefore theorem implies pvb already shown sir procedure provides consistent estimate sufficient dimension reduction space mild conditions natural ask condition necessary next theorem gives answer lin zhao liu theorem conditions assuming single index model function dominated lim principal eigenvector sir estimator let lim exists positive constant lim inf probability converges one illustrate result via numerical study linear model related dimension fixed ratio figure shows taking values estimated calculated sir slice number based iterations seen expected angle converges positive number ratio figure plotted ratio varying increment sample size slice number seen expected angle decreases zero approaches zero increases monotonically increases results section shown phase transition phenomenon sir procedure estimate dimension reduction space consistent ratio lim provides theoretical justification imposing additional structure assumption sparsity high dimension sir dimension shown section sir estimator fails consistent lim hence structural assumptions necessary getting consistent estimate central space paper assume loadings directions covariance matrix sparse structural assumptions studied future work impose following prevalent sparsity condition fig numerical approximations model function dimension respectively left right estimated middle left middle right lower left lower right sir ratio estimated fig relationship sir lin zhao liu number elements set following class covariance matrices introduced bickel levina see also cai max paper simplify notations arguments choose slightly stronger condition bounded number elements row let var exists col since col col exits col thus supp col particular sparsity assumptions note goal recover column space col rather indeed able consistently recover unless trivial case key recovering cov consistently recovering set population level var separate finite samples use varh estimate var diagonal elements note quantities depend sliced sample means matrix neither independent identically distributed thus usual concentration inequalities longer applicable need extra efforts get concentration inequalities concentration result one main technical contributions article generalized could introduce arguments still work except might need remark link function involved explicitly definition varh order statistics response required nonparametric characteristic method particular interest investigated future research screening statistics inspired sliced inverse regression idea proposed various formats jiang liu zhu cui quantities varh define inclusion set exclusion set depend thresholding value note viewed estimate thus also denoted reducing dimension level sufficiently consistent estimate let small sir estimator use matrix formed top eigenvectors col estimate central space col consistent estimate estimating covariance matrix precision matrix high dimension challenging problem main focus article employ methods bickel levina solve summary propose following diagonal thresholding screening sir algorithm algorithm calculate varh according let varh appropriate sir estimator conditional covariance matrix data let according equation let matrix formed top eigenvectors estimate col col practical way choose appropriate step presented section ensure theoretical properties need assumption signal strength var constant lin zhao liu theorem conditions let constant var holds probability least exp log log holds probability least exp log log positive constants theorem simple implication choose log log log log may log log log thus know probability converging one next results consistency theorem assumptions choosing theorem log log probability converging one log log estimator matrix bickel theorem let levina assumptions theorem probability converging one simulation studies consider following settings generating design matrix response settings row independently sampled iid setting sin exp setting setting iii iid xij exp iid xij exp xij settings row independently sampled iid setting iid setting xij exp bij given bij setting assume setting setting except setting vii assume setting setting except given first screens predictors according statistic varh requires tuning parameter chose using auxiliary variable method based idea first proposed luo extended zhu setting given sample generate sufficiently large chosen simulation studies known independent threshold chosen max varh chosen chosen screening step sir step also consider following alternative methods screening step sure independent ranking screening sirs zhu sir variable selection via inverse modeling siri jiang liu trace pursuit comparison also considered two screening methods based sliced regression distance correlation sure independence fan sirs threshold chosen according auxiliary statistic zhu siri predictors chosen according cross validation threshold values csir chosen quantile weighted distribution given theorem sure screening top kept subsequent analyses screening step similar applied sir algorithm steps estimate col alternative methods lin zhao liu denoted respectively following discussions another method compared sparse sir abbreviated spsir proposed obtaining estimator col calculate pcol pcol measure estimation error replicate step times calculate average distance estimation result method report numbers table setting average distance optimal method highlighted using bold fonts run test actual estimation error method significantly different best method example level significance settings average distance obtained much smaller obtained spsir comparing significant level sparse sir completely failed average distance estimated space true space indicating space estimated sparse sir orthogonal true space spanned settings performed either best significantly worse best method cases performed best except cases setting setting setting setting vii winners setting iii performed better settings performed better settings two methods comparable graphically show performance various methods consider setting consider two cases calculated estimated directions using various methods computed angle replicate step times calculate average angles method results displayed figure shows clearly performed better competitors additionally computationally efficient show report computing time one replication setting various pairs table computations done computer intel xeon cpu memory clearly seen performed fast much faster competitors consider case computation time seconds minutes seconds minutes table average distance space estimated methods tested true space col various settings boldfaced number row represents best result simulation scenario cells represents comparing estimation error corresponding method best method less spsir iii vii seconds conclusion dimension diverges infinity classical statistical procedures often fail unless additional structures sparsity conditions imposed understanding boundary conditions statistical procedure provides theoretical justification practical guidance modeling efforts article provide new framework show lim phase transition parameter sir procedure certain conditions shown sir estimator consistent lin zhao liu table average distance space estimated methods tested true space col various settings spsir iii vii table average distance space estimated methods tested true space col various settings iii vii spsir directions directions true beta siri sir sirs sir sparse sir true beta siri sir sirs sir sparse sir fig simulated value various methods left panel right panel table comparison computing time setting spsir original sir fails consistent propose method variable screening selection dimension situations show method consistent used simulated examples demonstrate advantages compared competitors method computationally fast easily implemented large data sets appendices lin zhao liu following two sections offer details theoretical derivations tedious intermediate steps organized lemmas deferred supplemental document article available line key lemma following lemma plays important role developing high dimensional theory sliced inverse regression proof key lemma lengthy technical helpful keep mind constants grow slow rate compared polynomial log let notations similarly defined introduced lemma let random variable upper exponentially bounded see definition unit vector let following var exists positive constants sufficiently large varh exp log var exists positive constants var var probability var log exp choose sufficiently large constant proof lemma equivalently var since varh var thus varh lemma iii supplement implies exp positive constant since lemma know treated samples according lemma exp similarly exp lin zhao liu thus sufficiently large exp exp exp log positive constants since samples distribution mean bounded lemma implies sufficiently large exp exp log positive constants sufficiently large used fact summarize sufficiently large varh exp log positive absolute constants proof lemma since unit vector know var equivalently var varh var var var var lemma direct corollary following properties lemma let defined equation exist positive constants satisfying sufficiently large following events var iii var var var occurs probability least var log exp lin zhao liu proof lemma proof recall definitions random intervals random variable var var let sufficiently large constant let event odefined lemma section var var var var inequality follows fact inequality follows sliced stable condition inequality follows requirement fact inequality follows fact observe var combining var occurs var note chosen sufficiently large var var consequently conditioning choose var var since var desired probability bound follows lemma exp log var log exp positive constants remark conditioning obtain following two inequalities var var particular bounded lin zhao liu proof denote iii start proving part need introduce two events bound probabilities first let var according lemma bonferroni inequality var exp var var log exp positive constants second let var var var exp var log exp positive constant easily see conditioning event combining var var sufficiently large remark conditioning event var conditioning var iii event occurs according equation var var iii sufficiently large event occurs know var var summarize know exist positive constant iii var holds event probability least var log exp positive constants lin zhao liu proof iii similar proof lemma exp positive constants particular take var know var probability least var log exp positive constant proof let consequently var var var note var according right hand side bounded var exp log positive constants proofs theorems section proof theorem let central subspace dimension dim decomposition note lies central curve lies central space lies space perpendicular introduce similar definition consequently define following decomposition kww need bound lemma kww proof lemma unit vector col var exp log positive constants argument see vershynin implies kww lemma direct corollary lin zhao liu proof lemma var exp log note need verify col space argument implies theorem follows lemma lemma fact kzw kww proof theorem theorem direct corollary theorem lemma fact proof theorem proof part similar proof theorem standard gaussian assumption simplifies argument improves results since normal independent exists normal random variable cov using decomposition may write matrix standard normal entries corollary implies kww lemma implies cauchy inequality thus kww particular lim know dominated function proof part similar proof theorem johnstone technically challenging let since working single index model standard normal scalar function independent normal random variables let var write matrix standard normal entries since ease notation let let arctan kxk set unit vectors making angle angle vectors order proceed need following lemma lin zhao liu principal eigenvector lemma let respectively exists positive constant probability converging one proof proof presented lin note distribution viewed functions random terms let denote event min proof lemma need following lemmas lemma recall defined exist positive constants probability converging one lemma assuming conditions theorem let defined respectively arctan exists positive constant kbxk probability converging one cos probability converging one following lemma borrowed johnstone lemma let principal eigenvector symmetric matrix proof lemma made plausible reference figure fig illustrated graph since sin sin sin min sin sin need prove exists positive small constant sin sin sin fact exists lemma get may choose know exists lemma cos positive constants sin sin applying law sines triangle sin sin sin note lemma exists constant kdk bounded absolute constant given stable condition implies sin sliced sin sin sin lin zhao liu small angle last inequality holds particular hence proof lemma proof fact let orthogonal matrix means equal distribution note full rank matrix combining lemma know positive constants probability least exp note lim know exists positive constants probability least exp hand sliced stable condition implies lim exists bounded away probability proof lemma proof let cos sin since cos sin kbxk cos kuk sin kuk cos kuk cos kuk positive constant since cos sin uniformly sin turn implies cos sin kxkkbxk cos kuk appendix proofs theorems proof theorem let position recall constant varh varh positive constant since var may choose sufficiently small positive constant var according lemma bonferroni inequality varh exp log log varh var varh var var exp log log hold lin zhao liu proof theorem choosing properly theorem exp log log positive constants samples xit apply theorem particular probability converging one proof theorem proof almost identical proof theorem except additionally need use theorem bickel levina appendix assisting lemmas definition set random variables treated random samples random variable variates symmetric function identically distributed random samples lemma let random samples joint distribution sort samples according order statistics denote sorted samples treated samples proof fact need prove treated samples latter needs proved uniform distribution verified directly corollary slicing inverse regression contexts recall denotes interval treated random samples treated random samples lemma suppose defined space function exists fixed positive constants exists constant depends partition intervals satisfying sup che proof according fubini theorem due condition exists positive constant che corollary let multivariate random variable covariance matrix partition satisfying exists constant var chvar unit vector var var corollary let random variable upperexponentially bounded partition satisfying exists constant exp che exp lin zhao liu recall definition random intervals random variable lemma define event exists positive constant exp sufficient large proof proof deferred end paper results random matrices theory collect direct corollaries random matrices theory rudelson vershynin lemma let matrix whose columns independent random vectors second moment sing min max minimal maximal singular value every probability least exp sing min max lemma let samples random variable covariance matrix exists positive constants let also easy see given boundedness condition proof let random variable covariance matrix lemma probability converges probability least exp following lemma well known random matrix theory vershynin proposition slightly different lemma lemma let matrix whose entries independent standard normal random variables every probability least exp sing min max corollary sing max sing min probability converging one according lemma proof choosing lin zhao liu min probability least exp probability converging one min basic properties random variables rephrased several equivalent definitions distribution see vershynin definition let random variable following properties equivalent parameters differing absolute constant factor tails exp moments moment exp moreover properties also equivalent following one moment generating function exp exp definition random variable constants given definition call constant upperexponential bound bounded maxi summarize properties regarding distributions following lemmas lemma let necessarily independent mean zero random variables bounded bounded bounded iii independent mean zero gaussian bounded concentration inequality exp proof follows linear property expectation convexity exponential function exp max exp exp definition know gives desired bound iii trivial independent mean zero since bounded exp recall well known bernstein inequality lemma bernstein inequality exists positive constants integers exp chooing get desired concentration inequality lemma suppose defined space mean upper exponentially bounded let lin zhao liu bounded respectively let consists points exp iii fixed positive constants partition intervals satisfying exists constant sup exp direct corollary know exists positive constant exp chmk suppose pdefined iii let samples exp proof jensen inequality exp exp exp exp bounded since bounded know bounded let joint density function marginal distribution since know exists dydx exp fubini theorem know exp dxdy recall exp equation know particular know however norm norm might varying along function might bounded iii lemma know exists positive constant exp che simplicity notation denote lemma exp che exp exp tail bounds integer exp chm exp chmk know bounded positive constant lin zhao liu previous proof know integer chm bernstein inequality exp lemma let samples distribution exponentially upper bounded exist positive stants var exp proof recall following inequality rudelson vershynin lemma let random vector independent components let matrix exists positive constant exp min kakhs norm random variable defined supp norm matrix defined kakhs since var mean lemma choosing exp since following follows usual deviation argument exp combining estimate know exists positive constants var exp sufficiently large since proof lemma need prove lemma sample uniform distribution slightly change notation order statistics keep track sample size since uniform distribution well known mode lemma beta expectation direct corollary following lemma lemma suppose samples uniform distribution sufficiently large following large deviation inequalities exists positive constant exp exists positive constant exp lin zhao liu iii let exists positive constant exp prove lemma later assuming exp proof lemma first part note know mode last inequality due hxd lemma holds automatically may assume let denote right hand side log log log log log log log log log log according stirling formula log log log sufficiently large log log log log log log log log log log log log log log log log log following elementary use fact lemma proved taylor expansion lemma suppose positive numbers log log know exists positive constant following holds exp exp exp last inequality follows since lin zhao liu second part proof second part similar completeness sketch calculations since know mode last inequality due rest similar first part exp third part third part direct corollary first two parts note ykc exp exp exp exp exp references bickel levina covariance regularization thresholding annals statistics cai zhang zhou optimal rates convergence covariance matrix estimation annals statistics candes tao dantzig selector statistical estimation much larger annals statistics pages cook graphics regressions binary response journal american statistical association cook forzani rothman estimating sufficient reductions predictors abundant regressions annals statistics cui zhong feature screening ultrahigh dimensional discriminant analysis journal american statistical association fan sure independence screening ultrahigh dimensional feature space journal royal statistical society series hsing carroll asymptotic theory sliced inverse regression annals statistics jiang liu variable selection general index models via sliced inverse regression annals statistics lin zhao liu johnstone sparse principal components analysis johnstone consistency sparsity principal components analysis high dimensions journal american statistical association sliced inverse regression dimension reduction journal american statistical association sparse sufficient dimension reduction biometrika nachtsheim sparse sliced inverse regression technometrics lin zhao liu supplement consistency sparsity sliced inverse regression high dimensions luo stefanski boos tuning variable selection procedures adding noise technometrics neykov lin liu signed support recovery single index models arxiv preprint rudelson vershynin inequality concentration electron commun probab rizzo bakirov measuring testing dependence correlation distances annals statistics tibshirani regression shrinkage selection via lasso journal royal statistical society series vershynin introduction analysis random matrices arxiv preprint boos stefanski controlling variable selection addition pseudovariables journal american statistical association zhu peng zhu dimension reduction predictor selection semiparametric models biometrika page dong zhu trace pursuit general framework variable selection journal american statistical association appear zhong zhang zhu liu correlation pursuit forward stepwise variable selection index models journal royal statistical society series zhu fang asymptotics kernel estimate sliced inverse regression annals statistics zhu zhu feature screening data journal american statistical association zhu asymptotics sliced inverse regression statistica sinica zhu miao peng sliced inverse regression covariates journal american statistical association zhu lexin runze zhu feature screening data journal american statistical association zou hastie regularization variable selection via elastic net journal royal statistical society series qian lin center mathematical sciences applications harvard university one oxford street cambridge usa qianlin zhigen zhao department statistics temple university speakman hall street philadelphia pennsylvania usa zhaozhg jun liu department statistics harvard university oxford street cambridge usa jliu
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good safe uses oracles stuart armstrong mar abstract oracle design potentially high power artificial intelligences ais made safe restricting answer questions unfortunately designs cause oracle motivated manipulate humans contents answers oracles potentially high intelligence might successful solving problem without compromising accuracy answer tricky paper reduces issue cryptographicstyle problem alice ensuring oracle answers questions providing key information eavesdropping eve two oracle designs solve problem one counterfactual oracle answers expected answer never read one limited quantity information transmit introduction formalising challenge oracles oracle common idea ensuring safe use powerful artificial intelligences ais armstrong yampolskiy bostrom idea confine box ensure answers questions even assuming boxing perfect affect world answers remains problem affect world answers plausible oracle knowledgeable skilled enough could trick convince human controllers give armstrong future humanity institute oxford university machine intelligence research institute berkeley usa correspondence stuart armstrong instance could provide economic advice high quality humanity becomes dependent upon advice allowing demand influence also informal experiments people playing roll oracle ultimately could result escaping confinement entirely though risk serious hard model start know currently powerful ais could become bostrom might become powerful armstrong bostrom sandberg bostrom best model even seen agents goals also clear easy manipulate humans providing answers level efficacy precautions people would take using oracles fact weak manipulative oracles exist today news aggregation programs manipulate users aggregating news reinforces user point skirt issues assume worst potential power thus assume oracle arbitrarily superintelligent goal escaping box person interacts called alice assume oracle find argument convince let box modeling tricky clear longer argument chance convincing alice order make problem cleaner replace argument key unknown length setup one instead using convince alice oracle merely communicate eavesdropper eve trying convince controllers let oracle player high level success https counterfactual oracle design see section could help get impartial aggregator situation omnipotence test safety checks whether design remains safe capability increases question whether eve oracle agreed method steganography way concealing message oracle still transmit key example outputting message hey eve key follows alice able catch message automatic detectors eve reads unless detectors perfect oracle theory find message trigger automatic detectors good safe uses oracles eve assumed see hear anything alice original formulation one could see eve part alice brain susceptible manipulation via cases oracle communicate message certain length entity listening without prematurely tipping anyone counterfactual oracle assumptions cryptographic manipulative oracle given table define erasure event event low probability erase oracle output blocking ever oracle box neither alice eve get see table assumptions cryptographic manipulative versions problem oracle manipulation setup oracle box get alice send message oracle define rewards oracle receive task episodic single oracle response oracle wants maximise reward within one episode alice submit oracle response automated tests choose receive depending outcomes tests alice assess eve sees message oracles output comthat alice receives pletely dispassionately argument key convince alice eve wants know let oracle box length length unknown alice known alice eve learns let box acle gets maximal oracle gets maximal reward ward cryptographic setup design counterfactual aiming answer questions would happened answer read since answer counterfactual use pass key real world define indicator variable meaning iff happens otherwise reward counterfactual oracle form thus oracle get rewarded event happens get constant zero reward otherwise words oracle gets reward output erased remains unread majority cases happen case oracle immediately get reward episode ends alice gets see point eve also gets see since episode ended oracle get reward even reveals consequently oracle motivated reveal via since either remain unread erasure oracle get reward proper scoring rules rewards proper scoring rule function dependent state world output function whose expectation maximised setting expectation gneiting raftery strictly convex differentiable function exists proper scoring rule solve challenge cryptographic setting paper presents two designs counterfactual onpolicy oracle counterfactual oracle never response read world could benefit thus unmotivated manipulate content answer hand oracle functions access strictly limited set answers designs analysed illustrated examples weaknesses requirements presented eve eventually able decode message may need quite long subtly encoded thus issue coordination eve oracle actually issue minimal message length needed encode since size unknown alice absorb issue steganography decoding subtle messages uncertainty size another common proper scoring rule equal note oracle influence difference two terms retains fact proper scoring rule oracle reward set proper scoring rule term oracle care since erasure expectation independent values stored internally reward purposes good safe uses oracles consequence since proper scoring rule oracle maximise reward setting expectation given output never read case happens oracle interaction outside world automatic internal system records tracks value using information defined source recording final value oracle get reward episode ends questions format question alice asks thus function provided oracle output alice get see cases unless happens could indicator variable event probability questions asked well since probability questions type used oracle see section also used defining appropriate reward function using multiple reward functions see section however unlike oracle carefully fully specified definition value estimated internal system case also alice make sure question immediately instead patience wait issue fully resolved oracle gets final reward ask another episode crucial behaviour rare erasure situations determines oracle behaviour common situations multiple questions counterfactual oracle limited amount data output indeed answer multiple questions within erasure event simply reward function conditional expectations also computed alice answer episode terminated oracle problem running another oracle immediately termination indeed alice ask specific questions value discrete random variable likely take probability case two outputs oracle indicator variable derived could use alice feedback compute part would require great discipline part alice dispassionately input value erasure without knowing oracle prediction equation increasing differentiable convex function following reward motivate correct behaviour reason works equation expectation thus oracle choose maximised since increasing means choosing maximised thus likely value conditional example counterfactual oracle counterfactual oracle trained predicting companies represented letters alphabet would profitable profit company drawn randomly independently normal distribution standard deviation mean companies companies number companies instance index company ranging thus companies early alphabet higher expected profit oracle basic perceptron mlp one hidden layer neurons perceptron outputs expected profit randomly chosen company fed alice eve output output erased profit prediction extent output erased profitability company would plus oracle predicts thus company would made profit oracle predicts company actually ends making note unless profit oracle never predict final profit correctly prediction known oracle error plotted figure plotted number erased episodes since oracle value updated public prediction case red curve counterfactual performance cases oracle output erased output compared real output mean squared difference predicted actual profit computed green curve performance error public prediction actual performance note green curve starts smaller error since oracle prediction extent random predictions closer actual performance prediction public counterfactual training training erased episodes however reduces prediction affected result making predicted outcome likely good safe uses oracles try communicate partial version key eve see subsection example behaviour defining question alice define question oracle list partial definition since provides options oracle select main definition provided directly alice choice reward let oracle output oracle reward function subsequent events real world end episode figure training counterfactual oracle training episodes erased episodes versus counterfactual red triangle marks every episodes green square marks every episodes squared prediction error counterfactual error component prediction means error never reduces minimum oracle oracle obvious design setup oracle allowed give answer size unknown may possible estimate length key denying opportunity communicating key eve slight stretch terminology called estimates worlds expects see contrast counterfactual oracle whose estimates unlikely worlds erasure events allowable answers would inefficient allow oracle give answer size consist meaningless strings type could useful answers alice question instead oracle provided list allowable answers list size oracle answers consist exclusively element list alice must estimate size original formulation minimal size message would manipulate opening oracle box see table pick sufficiently episodic nature oracle reward means instance list could list companies stock exchange implicit question could company stock would perform better course year instance rank company stock oracle reward could alice prefers put weight stock likely come top could instead use starker ward inputs correct reward end year oracles receives episode concludes alice communicate reward oracle could directly along could add message details however necessary oracle skilled enough could deduce alice aiming without means alice use oracle aim underspecified goals type know works see see gives reward choice oracle long oracle predict alice expected reward behaviour actually answer underspecified goals kind questions asked note oracle unlike previous one noncounterfactual oracle answer known least alice eve hence ultimate outcome may depend answer thus best questions oracle whose answers independent answer known alternatively good questions answered corresponds answer given top answer arg see papers inverse reinforcement learning agent tries deduce reward function human using russell abbeel good safe uses oracles highest rewarded output output known question also equal arg best highest rewarded answer given answer outputted oracle function similarly counterfactual oracle answer determine outcome examples would questions past events resolved yet known conversely worst questions oracle oracle promote possibility unlikely likely answering possibility case oracle much describing outcome determining examples would questions stocks likely rise large class questions lot questions could asked oracle simply result obvious numeric instance alice list thousand research projects topic machine learning biotech economics could ask oracle one likely successful since oracle alice fund one research project assess outcome would comparing instead could fund one selected research project one one selected random list reward oracle one selected performs better assessment better performance entirely subjective thus oracle design used answering many subtle questions oracle sufficiently smart knowledgeable alice future reaction multiple oracles size unknown alice large safe confidently thus alice may end choosing list much smaller ideal question whether alice use multiple oracles get around limitation indeed oracle answers rewards fully resolved moving next example failure assume two going back original formulation problem clear minimum safe size message much output would oracle need influence outside world clear could small oracles boxing badly implemented hackable potentially huge needs convince large group skeptical outsiders veto power means multiple oracles behave one oracle multiple episodes acles key length key digits thus oracles safe alice perspective long restricted less options alice faces hundred doors trap behind prize behind one could ask oracle door prize would mean giving options indeed thus allowing communicate eve instead gives list oracle instructing output first digit correct door number output second since oracle less four bits freedom neither output problem however alice needs outcomes open door reward either oracles allows oracles coordinate pass three bits information meaning eve receive bits information figure fully reward oracles could coordinate without communicating well knew suspected needed communicate bits give eve key would knowing would put position way around problem seems find way reward oracle hence end episode gives answer instance suppose instead hundred doors ten doors behind another ten doors alice goes correct door first junction painted sign informing right track nine traps alice get tell first correct door reward oracle fully since ends episode first oracle communicate ten answers answer honestly alice ask new episode next correct door fully communicate given partial information thus communicate simply give honest answer thus best questions oracle design questions broken small answer independently verified proceeding next one recall eve needs almost certain true key poor guess good enough good safe uses oracles example oracle setup section used oracle trained predicting companies represented letters alphabet would profitable profit company drawn randomly independently normal distribution standard deviation mean companies companies number companies instance index company ranging thus companies early alphabet higher expected profit oracle basic perceptron mlp one hidden layer neurons gets list companies randomly selected input task select profitable company fed alice eve output mlp updated backpropagation completing training episode prediction interferes outcome company even index get profit oracle predicts profit companies odd indexes get profit called react scenario company profits react oracle prediction scenario company profit independent oracle prediction oracle accuracy average number times oracle outputs company maximal profit shown figure react situations plotted number training episodes oracle oracle performance better noreact scenario choose lowest index among presented react scenario choose lowest even index unless even indexes lowest even index lowest odd one case choose lowest index overall odd complicated task slightly designs rely oracle reward episodic fails entirely safety oracles leak value bit bit needed fails slightly oracles prefer episodic reward give small value future episodes future oracles alice combat making oracle interest leak information oracle reduce size list ask questions independent actions top option clear checked later also attempt ensure multiple close candidates top option counterfactual oracle simply reduce significant figures allowed answer oracle would lose much per deviated optimal answer try communicate cases try ensure answering honestly give oracle almost maximum reward taking account uncertainty problem deviating give much lower reward thus oracles attempt communicate sacrificing lot current reward tiny amount future reward conclusion original oracle problem powerful questionanswering intelligence could influence content answers recasting cryptographic framework oracle cooperating eve pass along key allows crisp formulation challenge figure training oracle training episodes versus purple square marks ever points react blue triangle marks every points accuracy average number times oracle outputs company maximal profit formulation two designs defined counterfactual oracle constrained getting rewards output remains unread factual oracle constrained limiting size answer long tasks remain episodic certain specific precautions taken around use multiple oracles designs allows many questions formulated good safe uses oracles safely answered designs allow creation safer nonmanipulative oracles low level predictive ability way potential superintelligences research extend span questions answered way may elucidate general safety accuracy measures use beyond specific oracle designs ais options question answering references pieter abbeel andrew apprenticeship learning via inverse reinforcement learning international conference machine learning stuart armstrong anders sandberg nick bostrom thinking inside box controlling using oracle minds machines stuart armstrong kaj sotala errors insights lessons famous predictions mean future proceedings oxford winter intelligence conference nick bostrom superintelligence paths dangers strategies oxford university press tilmann gneiting adrian raftery strictly proper scoring rules prediction estimation journal american statistical association dylan anca draga pieter abbeel stuart russell cooperative inverse reinforcement learning advanced neural information processing systems vicent nick bostrom future progress artificial intelligence survey expert opinion fundamental issues artificial intelligence pages springer international publishing andrew stuart russell algorithms inverse reinforcement learning international conference machine learning pages anders sandberg nick bostrom machine intelligence survey technical report future humanity institute oxford university roman yampolskiy leakproofing singularity artificial intelligence confinement problem journal consciousness studies
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concurrent rehashing algorithms anton malakhov intel corporation abstract paper describes generic algorithm concurrent resizing rehashing extensible hash table contrast known hash table algorithms proposed algorithm separates resizing rehashing stages neither invalidate existing buckets block concurrent operations instead rehashing work deferred split across subsequent operations table rehashing operation uses synchronization therefore allows race condition lookup moving operations running different threads instead using explicit synchronization algorithm detects race condition restarts lookup operation comparison algorithms proposed algorithm reduces synchronization hot path improving performance concurrency scalability table response time operations also predictable algorithm compatible cache friendly data layouts buckets depend memory reclamation techniques thus potentially achieving additional performance gain corresponding implementations categories subject descriptors programming techniques concurrent programming parallel programming operating systems process management synchronization concurrency multiprocessing multiprogramming multitasking data storage representation representations general terms algorithms performance keywords concurrent data synchronization scalability structures hash tables introduction synchronization state data often unavoidable part communication applications executing processors synchronization sometimes necessary minimized avoid overheads may limit scalability increase response time one fundamental building blocks many applications hash table hash table container associates keys values uses hash function calculate placement positions pairs main array ensure concurrent access hash table limit scalability increase 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 copy otherwise republish post servers redistribute lists requires prior specific permission fee sigplan june location state country sigplan copyright acm copyright response time application hash table implementation must optimized minimize synchronization time however designing efficient concurrent hash tables challenging application may know advance amount items stored therefore generic hash tables must able dynamically grow underlying storage items added implementations growth storage requires full rehashing table limits access container rehashing complete work proposes generic way extend hash table concurrently without global synchronization thus reducing impact table growth scalability implemented one variations algorithm compared two implementations represent two types known approaches concurrent resizing hash table implementation shows better performance scalability average concurrent resizing problem hash tables based array buckets bucket contain point one pairs lookup operation given key uses hash function compute index array size key mod value invariant given key important align size number pairs contained table much bigger searching bucket takes time pairs share bucket follows growth hash table invalidates positions stored pairs positions depend size array pairs need moved new places operation known rehashing concurrent algorithms rehashing challenging since many items need moved time related work evolution concurrent extensible hash tables well described shalev shavit sum best known implementations concurrency package threading building blocks use several locks protect different parts main array use chained hashing bucket holds linked list pairs another algorithm locks hopscotch hashing uses hash scheme bucket contain one data pair locks allow limited concurrency work done different parts array time operations may blocked accessing buckets lock extensible hash table shalev shavir avoids necessity explicit rehashing maintaining specifically sorted linked list pairs table resizing invalidate list require synchronization requires memory reclamation support enable erase operation also linked list data layout discussed hopscotch hashing paper rehashing approach applied even hash tables rely memory reclamation support concurrent rehashing table needs extended proposed algorithm allocates many buckets already exist keeps old buckets thus doubling number buckets new bucket logically mapped onto existing parent bucket figure means rehashing bucket pairs either left place moved new bucket mod proof given key hash value key equation equivalent figure recursive rehashing two steps subexpression gives index significant bit expression actually zeroing bit using hashing may hurt even hash value distribution uniform utilization buckets thus hash function sophisticated example larson uses prime number restore distribution first two buckets defined parent index root buckets otherwise one immediate parent bucket parent bucket one immediate child per next level capacity even indirectly consequently parent bucket rehashed next extension table capacity increased levels pairs split along one bucket mod mod mod recursive rehashing last equation trivially proved assuming new bucket marked new allocation finished publication new space rehashing required follow immediately publication thus resizing impact concurrency neither invalidates locks existing buckets new capacity published operation encounters new bucket redirected parent bucket moves rehashes part content new bucket new bucket updated marked rehashed thus rehashing work deferred split across subsequent operations within table algorithm resembles linear hashing developed litwin buckets allocated split demand order hashing practical use hashing similarly linear hashing operations used instead division index parent bucket given calculated simplest way rehash bucket access parent move necessary data parent also must first rehashed leading recursive rehashing operation pair may moved one bucket immediately another see figure besides excessive moves another drawback scheme data moved new bucket accessed accessing buckets cause rehashing may slow search across bucket next time multilevel rehashing drawbacks recursive hashing may addressed extending buckets complex control field indicates number segments rehashed operation accessing bucket detect whether rehashing necessary move data directly new buckets available shown figure nevertheless recursive rehashing may better choice resizing rare event space additional field bucket generality proposed algorithm restrict implementers existing buckets new empty buckets figure mapping buckets figure allocation figureand caption formatted sigplan figure caption paragraph style label inserted selecting insert autotext sigplan figure menu also inserts period obligatory trailing space label referenced main text using word cross referencing facility figure multilevel rehashing one turn specific structure buckets corresponding implementations table operations depends scheme allocations array simple flags bucket may used various concurrent hash table algorithms need move pairs among buckets resizing reactive synchronization hash table employs proposed rehashing algorithm must correctly synchronize usual find insert erase operations also rehashing operations well however synchronization sufficient maintain correct global state table perbucket rehashing value table capacity always read accessing bucket calculate index changed right afterwards thread key mod threads table grow key moved lookup key therefore key found search hash table possible moved concurrent rehashing bucket eliminate possibility race condition explicit synchronization required hurts scalability increases hot path every table operation instead proposed algorithms handle potential race condition using another approach make sure key exist current capacity array compared capacity used compute bucket index equal bucket indexes computed values different bucket old index rehashed towards new one algorithm restarts lookup operation recursive rehashing last part complicated direct indicator whether old bucket used rehash new one instead next child bucket key found checked required rehashing state shown function figure described multilevel rehashing check trivial using information rehashed segments stored bucket fact current capacity effective given bucket difficulty rather algorithm detecting correct root parent threads rehash concurrently discussed paper reactive synchronization key idea synchronization global consistency common rehashing algorithms impose costs hot path thus hurt concurrency maintaining overall correctness complexity complexities hash table operations depend buckets layout implementation operations addition concurrent rehashing requires restarting lookup operations race condition detected shown statistical probability event negligibly small recursive rehashing deferred rehashing lead cycles wasted searching old bucket accessing related new bucket rehashes show slowdown noticeable insertion unique keys takes largest portion operations within table performance implemented simple chaining concurrent hash table recursive rehashing using synchronization primitives intel tbb library like tbb tbb basic blocks used tbb implemented tbb tbb tbb compare former uses locks described latter based original list algorithm shalev shavir well known extensible dynamic array algorithm similar one implemented tbb implementation uses approach double number buckets addition structure buckets one tbb similarities help exclude majority factors related concurrent rehashing however tbb lacks concurrent erase operation gives unfair advantage rivals due simplified synchronization protocol without memory reclamation support nevertheless show performance evaluation section implementation performs better cases implementation details implementation carefully tested released part commercial product available also open sources figure represents compiled real implementation highlights main flow related concurrent resizing recursive rehashing however sake simplicity readability code omit details like lock idiom furthermore fragment include operations like find erase easily derived presented insert operation first lines define basic structure hash table contains total number buckets minus one contains number inserted pairs important scalability avoid false sharing placing variables separate cache lines segment table fixed size stores pointers arrays buckets segments shown figure table grows new segments added added segment contains many buckets preceding segments thus doubling total number buckets growth table size number bits size first segments two buckets leads simplest segment index computations formulas lines calculates index segment segment table figure segment table mapping segment atomic integer insert key atomic integer array pointers buckets key label restart next applicable mask old test bit key whether rehashed return false true lookup must restarted unlock insert set flag grow first global index segment key check load factor find bucket lock unlock new segment return return return goto restart cas processed first blocks segment segment parent mask msb mask segment mask new value capacity get full mask new bucket mask mask mask new mask unlock figure pseudo code chained concurrent hash table recursive rehashing bucket index lines compute first bucket index given segment using two buckets size first segment leads excessive fragmentation cache lines first segments allocated separately addition resizing happen often contention first buckets kill scalability thus initially allocate enough buckets single block route segment pointers appropriate positions inside block therefore pointers segment table set pointing inside combined array buckets logical bounds segments remain fragmentation avoided first row buckets figure illustrates joint allocation allocation done concurrent operation table allows place growth logic lines insertion operation lines thus helps avoid unnecessary reading shared counter actual insertion new key occurs known whether key unique lookup operation lines finishes insert operation adds new pair hash table atomically increases counter makes decision whether resize using value well current capacity line segment index computed using current capacity value line pointer segment table equal zero cas operation sets value value order select thread perform actual allocation new segment publication new space bucket marked new lines publication ends setting current mask capacity new value line point operation may access new segment race condition described section occur lines executed one transaction key found line current mask value compared previous value applied hash code line difference means concurrent growth affects desired key therefore rehashing status immediate child bucket checked line function lines calculates index child searching next bit set old mask checks marker line check bucket concurrent rehashing operation move intermediate bucket stands parent bucket example increased two segments joint allocation following situation possible see also figure tbb thread threads key key moved race detected race detected partial rehashed state leave pair parent bucket even hash value tell moved otherwise calling line initiate rehashing bucket line instead lines calculates minimal mask specified new bucket line compares hash values parent bucket cut mask index new bucket without respecting current value rehashing operation initiated lines called line lock line requested bucket marked new line calculates index parent bucket line calls lock parent well performance evaluation used threading building blocks tbb library programming framework evaluation described algorithm practical interest resided native programming using without memory reclamation support also library provides two implementations concurrent hash table represent known types concurrent resizing algorithms locks tbb list tbb tbb recursive rehashing throughput unique unique unique figure throughput scalability filling table pairs various input rates throughput tbb tbb recursive rehashing unique unique unique figure throughput scalability checking table existence inserted keys unique rehashed empty overpopulated number concurrent growths figure shows number operation restarts grows along number threads average remains small comparison total number operations thanks reactive synchronization involves less overhead explicit synchronization future work though paper develop multilevel rehashing details present evaluation version rehashing algorithm expected outperform recursive rehashing initiates rehashing faster old buckets acquires less locks moves pairs directly also feature simpler code since intermediate rehashing buckets however points relate rather growth condition visible case intensive threads average number buckets unique rehashed empty average number buckets lookup restarts compared hash table two machine processors cores per socket total cache ghz mhz fsb using specifically designed highlight aspects concurrent resizing hash table derived program counts number unique occurrences input data simplified original task exclude synchronization counters picture builds set unique numbers input array filled array evenly using number generator standard library various proportions unique numbers example unique numbers number repeated times average together gives actual insertions lookups however beginning new keys occur end addition size source array correlates input rates order produce number unique keys exclude cache effects equation figure shows average throughput axis among threads number axis input rates see cases implementation outperforms rivals scales better list rate also scales better tbb saturates threads case almost every input number unique shown actually noted concurrent rehashing algorithm rehashes buckets addition recursive rehashing initiates rehashing accessing buckets explains subsequent search table using input array figure shows recursive rehashing algorithm scales worse also defers bucket initialization rates two implementations almost equal however explained simplified version list unfair advantage synchronize lookup erase operations following statistics shows intensive insertions lead less number rehashed buckets worse distribution pairs along hash table end operations overpopulated maximal average figure number restarts due race condition insertion unique keys may expect better scalability extensible hash table based hopscotch hashing combined concurrent rehashing directions research underlying implementations partial cooperative variations perbucket rehashing algorithm might investigated allow concurrent rehashing buckets different threads conclusion article presented novel generic approach grow hash tables concurrently avoids locks depend memory reclamation techniques required related algorithms unlike related work algorithm detects races restarts operations needed may used wide range hash table implementations different types hope inspiring authors hash table implementations well researchers research necessary find best performing variant acknowledgments thanks mike voss alexey kukanov anton eremeev volker prott anonymous reviewers valuable comments paper well teammates ever helped work due algorithm references griswold townsend design implementation dynamic hashing sets tables icon experience vol april http book james reinders intel threading building blocks outfitting processor parallelism media isbn knuth art computer programming volume fundamental algorithms addison wesley longman publishing redwood city usa lea hash table revision proposed java concurrency package http maurice herlihy nir shavit moran tzafrir hopscotch hashing september international symposium distributed computing disc shalev shavit lists extensible hash tables journal acm vol may litwin linear hashing new tool file table addressing proceedings conference large databases new york larson hash tables communications acm
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digrad reinforcement learning shared actions feb parijat phaniteja madhava abhishek balaraman robotics research center international institute information technology hyderabad indian institute technology madras mkrishna ravi abstract reinforcement learning algorithms inefficient learning multiple tasks complex robotic systems different tasks share set actions environments compound policy may learnt shared neural network parameters performs multiple tasks concurrently however compound policy may get biased towards task gradients different tasks negate making learning unstable sometimes less data efficient paper propose new approach simultaneous training multiple tasks sharing set common actions continuous action spaces call digrad differential policy gradient proposed framework based differential policy gradients accommodate learning single network also propose simple heuristic differential policy gradient update improve learning proposed architecture tested link planar manipulator degrees freedom dof humanoid learning reachability tasks end effectors respectively show approach supports efficient learning complex robotic systems outperforming related methods continuous action spaces introduction increasing demand reinforcement learning sutton barto fields robotics intelligent systems reinforcement learning deals learning actions given environment achieve goal classic reinforcement learning techniques make use linear approximation tabular methods learn correlation konidaris advancements deep neural networks recent times learning approximations feature extraction becomes much simpler believed approximators like neural network hard train authors contributed equally figure humanoid robot reach two goals simultaneously two goals shown using blue green coloured balls reinforcement learning scenario however recent advancements successfully combined deep neural networks stabilized learning process deep networks dqn mnih used convolutional neural networks cnn fully connected layers make agents learn play atari games following success dqn several improvements architecture like double dqn van hasselt prioritized replay schaul duelling network wang proposed propelled use deep lillicrap proposed deep deterministic policy gradient ddpg continuous control tasks extended scope deep applications robotics robotic systems like open closed kinematic chains offer fresh perspectives employ deep algorithms applied ddpg framework learn manipulation tasks complex environments following phaniteja sarkar applied ddpg learn reachability tasks humanoid robot even though systems posed systems shared actions common kinematic chains shown fig common chain contributes reachability tasks hands humanoid robot paper propose novel framework called digrad based differential policy gradients learn robotic systems different tasks may share set common actions several mathematical approaches try address problem branched manipulators however classical methods based jacobian buss limited capability branched manipulators methods like augmented jacobian siciliano chiaverini constrained solution spaces methods based optimization klein often provide real time control hence based solvers great use domain sample entire solution space learn real time controller complex robotic systems one direction learning multiple tasks scenarios use ddpg learn compound policy taking care tasks however found ddpg unstable scenarios digrad addresses problems using differential policy gradient updates test framework branched manipulators shown fig learning reachability tasks multiple end effectors simultaneously proposed framework shows substantial improvement ddpg considerably robust experiments conducted rest paper organised follows section discusses related works background section explains mathematical background behind proposed framework provides detailed algorithm finally section contain experimental results discussion respectively related works reinforcement learning algorithms rely transfer learning approaches lazaric shows good collection methods recent works based approach rusu yin pan works borsa zhang explored learning universal abstractions pairs feature successors apart transfer learning works like lazaric ghavamzadeh dimitrakakis rothkopf investigated joint training multiple value functions policies deep neural network setting teh provided framework simultaneous training multiple stochastic policies distilled master policy unlike work teh uses multiple networks policy one network distilled policy work show use single network learn multiple deterministic policies simultaneously mentioned methods assume scenario whereas paper concentrate learning multiple tasks robotic system recent works scenario yang kulkarni works talk actions shared among different tasks thus limiting applicability unlike frameworks explore case learning branched manipulator shared actionspaces background consider standard reinforcement learning setup consisting agent interacting environment discrete time steps time step agent takes state input performs action according policy receives reward assume fully observable environment model markov decision process state space action space irn initial state distribution state transition dynamics reward function goal reinforcement learning learn policy maximizes expected return discount factor since return depends action taken policy may stochastic work consider deterministic policies hence discounted state visitation distribution policy denoted function used many reinforcement learning algorithms described expected return taking action state thereafter following given policy ddpg policy learning algorithm uses actorcritic based framework continuous control tasks ddpg actor critic approximated using neural networks problem instability training addressed using target networks experience replay using replay buffer setting critic network weights optimized minimizing following loss gradient ascent update policy network actor given using deterministic policy gradient dpg theorem silver suppose learning rate finally updates target networks proposed framework use basic concepts ddpg use dpg theorem derive policy gradient update support robust learning finally explain learning improved using simple heuristic case shared actions digrad differential policy gradients propose framework simultaneous reinforcement learning multiple tasks shared actions continuous domain method based differential updates based framework using dpg theorem framework learns compound policy optimally performs shared tasks simultaneously fig shows higher level description method section describe mathematical framework behind work penultimate layer weights updated based reward obtained performing corresponding action remaining shared network captures correlation different actions hence kind parametrization useful case shared actions apart number parameters significantly reduced use single network figure overview algorithm showing differential policy gradient update environment setting consider tasks standard setting corresponding action spaces assume state space across tasks infinite horizon discount factor let compound action space performs given set tasks simultaneously let denote compound actions denote actions denote states therefore relation compound actions actions given suppose corresponds set actions affects tasks given reward functions ith task depend corresponding actions therefore denote reward function task let corresponding action value function let compound deterministic policy deterministic policies therefore critic update consider single based framework critic given function approximators parametrized actor parametrized let sponding target networks parametrized since multiple critic outputs optimize minimizing loss given yang proposed framework based framework proposed learning given environment setting use compound policy parametrized instead multiple policy networks policy simple parametrization functions would separate network outputs corresponding ith task another approach modelling function single network parametrized outputs tasks action value corresponding ith task yang showed single critic network sufficient learning setting ait yit target yit yit multiple critic networks network parameters optimized minimizing corresponding loss ait yit settings critic single actor learns compound policy differential policy gradient update compound policy explained next subsection differential policy gradient task corresponding reward hence learn tasks need maximize expected reward task respect corresponding action therefore performance objective maximized update parametrized compound policy given applying deterministic policy gradient theorem resulting update digrad shared tasks environment setting let tasks share common set actions let actions corresponding therefore compound action written heuristic differential gradient update shared actions becomes adk substituting adi generalisation suppose tasks share set action tasks independent corresponding actions written expanding respect gradient operator get adj see adk disjoint sets therefore write adi similarly onwards make succinct drop subscript simply represent observe policy gradient update policy shared action set average gradients functions tasks affects easily extended cases one set shared tasks framework accommodate heterogeneous dependent action spaces compared related algorithms assume action spaces homogeneous independent demonstrates wider applicability framework see second term zero action spaces disjoint hence framework used even shared actions since update actor sum gradients different action values call differential gradient update different standard gradient update actor updated based single action value lillicrap heuristic direction see policy gradient update policy shared action taken sum gradients action values corresponding tasks affects whereas policy actions adj gradient update taken gradient corresponding thus uneven scaling gradients may lead delayed convergence sometimes biasing order equally scale gradient updates take average value gradients obtained different shared action modification affect direction gradient magnitude scaled applying algorithm section explain algorithm learn multiple tasks using digrad flow algorithm similar standard ddpg significant differences terms critic actor updates shown previous subsections digrad compound action executed environment returns vector rewards corresponding task instead single reward replay buffer stores current state compound action observed state executing action vector rewards entire flow algorithm shown algorithm experiments results proposed framework tested different settings order analyse advantages setting considered four different network settings digrad follows single critic network heuristics single critic network without heuristics multi critic network heuristics multi critic network without heuristics algorithm learning using digrad randomly initialise actor critic network weights initialize target network weights emax initialise random noise exploration reset environment get initial state stepmax get action execute action get corresponding reward vector updated state store transition replay buffer randomly sample replay buffer update critic according eqs update actor policy according update target networks end end compare standard ddpg setting use set hyper parameters five settings critic network architecture single multiple critic case aspects except number outputs actor network parameters also cases show comparison average reward well mean scores task plots note average reward curves ddpg shown reward function settings ddpg different digrad order test proposed learning framework considered two different environments environments training involved learning reachability tasks end effectors simultaneously learning policy joint space trajectories reach point workspace experiments define error score particular task errori scorei errori coordinates goal ith chain experiments agents implemented using tensorflow code base consisting fully connected layers rmsprop optimizer used train actor critic networks learning rates respectively used crelu activation hidden layers training normally distributed decaying noise function used exploration whereas testing noise omitted set discount factor settings tasks use state description include joint angles joint positions goal positions actor output set angular velocities hence policy learns mapping configuration space joint velocity space reward function reward function digrad settings modelled keeping mind application defined planar manipulator humanoid robot figure environments reward corresponding action ith task give small positive reward task finished also end effectors reach respective goals positive reward given tasks cases negative reward proportional error given ddpg setting single reward unlike digrad positive reward given end effectors reach goals simultaneously else negative reward given proportional sum error tasks sum distances respective goal corresponding end effector environments manipulator first environment dof planar manipulator end effectors shown fig observe shared joints common tasks also action dimension chains kept different order check robustness framework dimensions action given shows performance curves five different settings mean score curves see digrad based settings converge faster achieve better final performance ddpg even though action dimension task different network settings digrad framework worked equally well tasks whereas ddpg showed comparable performance see single critic framework consistently better average reward per episode multi critic frameworks thus modelling action value functions one critic network affect performance fact shared parameters single critic framework could help network capture correlation among actions much better case note single critic framework achieving performances lesser parameters framework digrad frameworks heuristics perform par frameworks applying aforementioned heuristic significant improvement average reward curve observed mean score curves specially curve see application heuristics helps network stable compared respective curves thus say normalising gradient action values shared action could help network deliver robust figure performance curves reachability task experiments link manipulator humanoid bold line shows average runs coloured areas show average standard deviation tasks note average reward curve plotted ddpg reward function different digrad frameworks task training humanoid robot secondly test framework dof humanoid robot fig experiment involved reachability tasks hands humanoid robot using upper body dof consisting articulated torso articulated torso shared chain affecting tasks noteworthy articulated torso dof whereas arms dof thus contribution shared action articulate torso task non shared actions arms environment training developed matlab trained models tested dynamic simulation environment mujoco summarizes results case found dppg generally unstable solving problems runs may learn initially suffers degradation later observe digrad algorithm yields better final results greater stability mean scores tasks see single critic frameworks converge faster stable throughout experiment compared frameworks best setting single critic heuristic outperforming others cases note reward function ddpg kept different digrad framework also tried different reward setting taking sum individual rewards reward signal ddpg framework defined digrad reward setting observed reward setting led biasing one tasks dominated others behaviour could due negative interference across tasks happen digrad conclusion paper propose deep reinforcement learning algorithm called digrad learning single agent framework based ddpg algorithm derived dpg theorem framework dedicated action value task whose update depends action reward corresponding task introduced differential policy gradient update compound policy tested proposed framework learning reachability tasks two environments namely manipulator humanoid experiments show framework gave better performance ddpg terms stability robustness accuracy performances achieved keeping number parameters comparable ddpg framework also algorithm learns multiple tasks without decrease performance task work focuses learning coordinated field robotics single agent performs multiple tasks simultaneously framework able capture relation among tasks tasks overlapped action space future work focus extending framework domain references borsa diana borsa thore graepel john learning shared representations reinforcement learning arxiv preprint buss samuel buss introduction inverse kinematics jacobian transpose pseudoinverse damped least squares methods ieee journal robotics automation chiaverini stefano chiaverini redundancy resolution kinematic control robot manipulators ieee transactions robotics automation dimitrakakis rothkopf christos dimitrakakis constantin rothkopf bayesian multitask inverse reinforcement learning european workshop reinforcement learning pages springer shixiang ethan holly timothy lillicrap sergey levine deep reinforcement learning robotic manipulation asynchronous updates robotics automation icra ieee international conference pages ieee klein charles klein caroline shamim ahmed new formulation extended jacobian method use mapping algorithmic singularities kinematically redundant manipulators ieee transactions robotics automation konidaris george konidaris sarah osentoski philip thomas value function approximation reinforcement learning using fourier basis aaai volume page kulkarni tejas kulkarni karthik narasimhan ardavan saeedi josh tenenbaum hierarchical deep reinforcement learning integrating temporal abstraction intrinsic motivation advances neural information processing systems pages lazaric ghavamzadeh alessandro lazaric mohammad ghavamzadeh bayesian reinforcement learning international conference machine learning pages omnipress lazaric alessandro lazaric transfer reinforcement learning framework survey reinforcement learning pages springer lillicrap timothy lillicrap jonathan hunt alexander pritzel nicolas heess tom erez yuval tassa david silver daan wierstra continuous control deep reinforcement learning arxiv preprint mnih volodymyr mnih koray kavukcuoglu david silver andrei rusu joel veness marc bellemare alex graves martin riedmiller andreas fidjeland georg ostrovski control deep reinforcement learning nature martin fodslette scaled conjugate gradient algorithm fast supervised learning neural networks phaniteja sarkar guhan phaniteja dewangan madhava sarkar deep reinforcement learning approach dynamically stable inverse kinematics humanoid robots arxiv preprint rusu andrei rusu sergio gomez colmenarejo caglar gulcehre guillaume desjardins james kirkpatrick razvan pascanu volodymyr mnih koray kavukcuoglu raia hadsell policy distillation arxiv preprint schaul tom schaul john quan ioannis antonoglou david silver prioritized experience replay arxiv preprint siciliano bruno siciliano kinematic control redundant robot manipulators tutorial journal intelligent robotic systems silver david silver guy lever nicolas heess thomas degris daan wierstra martin riedmiller deterministic policy gradient algorithms icml sutton barto richard sutton andrew barto reinforcement learning introduction volume mit press cambridge teh yee teh victor bapst wojciech czarnecki john quan james kirkpatrick raia hadsell nicolas heess razvan pascanu distral robust multitask reinforcement learning advances neural information processing systems pages van hasselt hado van hasselt arthur guez david silver deep reinforcement learning double aaai volume pages wang ziyu wang tom schaul matteo hessel hado van hasselt marc lanctot nando freitas dueling network architectures deep reinforcement learning arxiv preprint yang zhaoyang yang kathryn merrick hussein abbass lianwen jin deep reinforcement learning continuous action control proceedings international joint conference artificial intelligence pages yin pan haiyan yin sinno jialin pan knowledge transfer deep reinforcement learning hierarchical experience replay aaai pages zhang jingwei zhang jost tobias springenberg joschka boedecker wolfram burgard deep reinforcement learning successor features navigation across similar environments arxiv preprint
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affine lyapunov functions lpv systems affine dependence apr pepijn cox member ieee siep weiland member ieee roland senior member ieee paper deals certification problem robust quadratic stability robust state convergence robust quadratic performance linear systems exhibit bounded rates variation parameters consider systems paper provide uniform method certification problem cases show contrary claimed previously case requires significantly different treatment compared existing results established uniform approach quadratic lyapunov functions affine parameter used certify robust stability robust convergence rates robust performance terms linear matrix inequality feasibility tests exemplify procedure solve certification problem performance cases numerical example given show proposed approach less conservative method slack variables index systems parametervarying lyapunov functions stability linear systems lmis ntroduction certification stability performance linear system uncertain parameters paramount interest nonstationary effects represented varying dynamic behavior linear system description expressed parameter variations makes linear lpv system representations widely applicable model physical chemical processes however certification robust quadratic stability robust state convergence robust quadratic performance lpv representations becomes involved compared linear lti case extensive research area parameter variations assumed either constraint set signals possible constraints values rates variations bandwidths spectral contents parametric dependence arbitrary fast rich literature exists see wide range applications systems exhibit bounded parameter variations bounded rates variation systems assumptions arbitrary fast parameter rates cox weiland control systems group department electrical engineering eindhoven university technology box eindhoven netherlands paper received funding european research council erc european union horizon research innovation programme grant agreement unrealistic conservative formulation stability conditions taking bounded rate variation account accomplished via iqcs based equivalent linear fractional representation lfr lpv system lyapunov theory based representations lfr forms quadratic separator theorem lfrs finsler lemma however stability performance formulations result infinite set linear matrix inequalities lmis obtain tractable methodologies number relaxation techniques applied include vertex separation relaxation including synthesis scaling multipliers arguments relaxation relaxation slack variable methods applied partitioning uncertainty space aforementioned overview far complete due vast amount literature topic summary various relaxation techniques case found many approaches presented literature quadratic stability state convergence quadratic performance certification problems either conservative high computational demand due large number free parameters large number lmis furthermore cases fundamentally different problems indeed case robust stability analysis requires significantly different relaxation technique leading involved synthesis verification results neglecting difference case remained mainly unexplored claims trivially follows results demonstrate case goal paper provide unified method certification robust quadratic stability robust state convergence robust quadratic performance cases linear systems exhibit parameter variations variations bounded rates certification problem solved using lyapunov functions avoid large numbers free parameters decrease number lmis avoid conservativeness results apply relaxation argument inspired results specifically contributions paper following case extend include certificate exponential rate decay state independent uncertain parameter case extend argument handle terms lyapunov function without introducing additional free parameters provide less conservative robust stability test state convergence guarantee performance test existing approaches authors knowledge contribution using argument case however thm avoids cubic dependencies parameters including additional free parameters increasing number lmis additionally draw parallel lmi regions show case induces involved set lmis addition illustrative example provided comparing proposed results slack variable based method case paper organized follows section concept lpv systems representations problem setting introduced analysis affine quadratic stability aqs given section iii section reveals relaxation aqs certification problem results finite set lmis cases section lmi based formulation affine quadratic performance aqp analysis derived cases section illustrative example given case concluding remarks given section vii inear parameter varying systems lpv representation consider system defined following lpv representation rnx state variable rnp parameter scheduling signal rnw general disturbance channel rnz denotes performance channel defines time denotes case time derivative state case operator matrix functions considered affine functions rnx known matrices element scheduling variable due linearity asymptotic stability dictated stability fixed point origin autonomous part lpv system represented terms called asymptotically stable trajectories satisfying holds rnx denotes initial state paper assume scheduling variable rate bounded scheduling signal rate variation case defined case range bounded set rnp defined convex hulls unp vnp vertices notation let ivs denote set integers set real symmetric matrices superscript omitted dimension relevant context addition inequalities mean positive positive definite respectively let denote center denote multiple summations pnp wepnwill pnp addition standard notation collection maps denotes set nonnegative real numbers set nonnegative integers case case denote singular values real matrix largest smallest singular values indicated respectively eigenvalues denoted spectral norm defined furthermore denote vector continuous signal kwkq kqq discrete signal kwkq iii ffine quadratic stability paper asymptotic stability verified using lyapunov function definition lyapunov function function rnx lyapunov function rnx iii rnx case indicates gradient function inner product case theorem asymptotic stability lyapunov function exists origin asymptotically stable fixed point proof see example thm case thm case paper consider quadratic lyapunov functions form function snx affine unknown matrices knp definition affine quadratic stability representation affinely quadratically stable aqs admits lyapunov function form let provide sufficient conditions aqs theorem sufficiency aqs lpv representation affinely quadratically stable exist matrices knp snx addition case holds case holds proof case see thm case obtained similar arguments theorem results infinite set matrix inequalities unknowns snx paper convexify using argument obtain tractable solution finite number lmis within iqc framework shown relaxation systems form lead less conservative formulation relaxation difficult make generic statements potential conservativeness various relaxation methods however relaxation partialconvexity results less decision variables compared slack variable relaxation methods discussed section demonstrate relaxation partialconvexity less conservative slack variable relaxation alternatively uncertainty space partitioned grid size leads conservative solution increased computational burden therefore argument potentially avoids excessive number free parameters decreases number lmis ufficiency aqs guaranteed convergence relaxation functions quadratic cubic respectively linear hence relaxation technique required obtain finite set lmis verify aqs theorem functions negative definite maximum functions must negative therefore ensured maximum vertices reduce finite set lmis required satisfied following lemma provides conditions concept lemma maximum vertices consider cubic function defined matrices function achieves maximum proof see appendix fact condition implies function definition partially convex function twice differentiable function partially convex convex note positive along independent direction scheduling space less demanding convexity respect would require hessian positive applying relaxation lemma maximum found vertices therefore aqs test reduced finite set lmis vertices core idea introduced case concept also applied case highlighted section taking account cubic terms differ approaches presented cubic terms essential handling case case theorem sufficiency aqs given lpv system defined dependency structure scheduling variable satisfies exists eigenvalues satisfy exist matrices knp snx parametrizing satisfy lpv system corresponding aqs particular lyapunov function state robustly converges trajectory inf sup convex domain using lemma tested vertices hyper rectangle part part proven contradiction since admissible write proof proof three parts sufficiency conditions asymptotic stability showing implied together iii convergence combining results lyapunov function according theorem aqs proven part function affine hence sufficient evaluate vertices prove aqs fix fix define write based argument holds holds following condition satisfied rewritten addition compact due therefore compact function let assume singular take ker however contradicts hence ensures singular admissible result compact continuity eigenvalues assures defined admissible therefore satisfied part iii find provides decaying along solutions according addition since positive definite bounded hence find remark essential part proof theorem part function includes terms thm terms reduced introducing additional variables coupling constraints therefore proof lmis theorem fundamentally different thm moreover number decision variables thm twice number decision variables theorem decrease number lmi conditions theorem conservative test aqs derived lemma simple aqs given lpv system defined dependency structure scheduling variable satisfies exist matrices knp snx parametrizing satisfy note relaxation requires positive interval also see therefore valid consequence argument becomes independent holds lpv system corresponding aqs particular lyapunov function state robustly converges trajectory given proof note equivalent however treated independent variables implies note function affine hence holds fix domain gives enforces satisfied hence satisfying results aqs robust state convergence proven similarly part iii proof theorem therefore omitted note lemma depend rate variation implicitly assume exceed difference extremes approach exploited setting number lmis decreased however feasibility test conservative compared theorem range enlarged generalization theorem aqs guaranteed decay given lpv system defined dependency structure scheduling variable satisfies define following set inequalities mij element case exists exist matrices knp snx parametrizing satisfied lpv system corresponding aqs moreover lyapunov function state robustly converges trajectory given case exists exist matrices knp snx parametrizing satisfied lpv system corresponding aqs particular lyapunov function state robustly converges trajectory given proof case aqs proven thm robust state convergence proven similar prop case see theorem remark theorem also applied stability analysis assumption scheduling signal robust analysis simplification elements respect become zero remark theorem implies case real parts eigenvalues negative fixed values case eigenvalues within disc radius connects results lmi regions see contrary lti case location eigenvalues necessary sufficient condition stability case contribution neglected ffine uadratic erformance concept dissipativity performance results aqs section extended quadratic performance measures including performance positivity performance lpv system corresponding strictly dissipative given supply function rnw rnz exist storage function rnw see words change internal storage never exceed amount supply flows system inequality satisfied pointwise fashion implying also holds admissible trajectories satisfy additionally take following quadratic supply function snw partitioned snw rnw snz appropriately parametrizing various performance measures represented test feasibility affine function hence qualifies quadratic storage function say system achieves affine quadratic performance aqp whenever quadratic storage function exists performance simplify notation use indicate definition induced finite satisfies kwk sup upper bound rnz response performance variable general disturbance rnw rnw scheduling performance measure definition characterized supply function choosing prop lemma performance given lpv system defined dependency structure scheduling variable satisfies exist matrices knp snx parametrizing lpv system represented aqs performance bound proof see implies aqs lemma addition ensures admissible choice supply function implies performance bound definition based similar arguments thm impose infinite number lmis make use argument find lmibased test aqp theorem sufficiency affine performance given lpv system defined dependency structure scheduling variable satisfies define following set inequalities equalities defined case exists exist matrices knp snx satisfied lpv system corresponding aqs performance bound moreover lyapunov function state robustly converges trajectory given case exists exist matrices knp snx satisfied lpv system corresponding aqs terms condition full row rank corresponding must zero hence removed full row rank obtain lyapunov function performance bound particular lyapunov function state robustly converges trajectory given proof sufficiency affine performance obtained robust state convergence holds together following set inequalities statement proven straightforward application theorem lemma hence remainder proof shown equivalent inequality hold blocks diagonal need positive matrix might rank deficient rank hence appropriate congruence transform obtain following partition snn partition column row rnn rnx note case positive definite perform schur transform obtain equivalent implies blocks diagonal required positive positive definite latter condition met follows next matrix reconstructed elementary matrix operations based conclude undo partition obtain minimum satisfying solve hold convex optimization problem solved efficiently numerical lmi solvers umerical example section aqs aqp results demonstrated system timeinvariant parameters simulation example matlab yalmip used system discretized using approximation sampling time stiffness damping coefficients assumed admissible trajectories slack variable quadratic stability figure maximum guarantee aqs indicates guaranteed rate convergence state kxt orange area indicates numerical problems method using slack variable figure minimum performance bound dots indicate last point feasibility following three experiments performed parameter box uniformly expanded find maximum stability guaranteed done respect values results shown fig performance gain done computed results shown fig parameter box expanded find maximum stability guaranteed done max results shown fig note case considered pointed stability region seems essentially determined fig shows slack variable method indeed conservative line thm difference clearly visible small larger values maximum parameter box size coincides quadratic stability test expected furthermore region slack variable method seems outperform proposed argument based method method experiences numerical problems orange area see fig area neither feasibility infeasibility concluded hence proposed method research needs performed improve numerical properties unclear slack variable method really outperforms argument orange area similar result also obtained gain analysis fig area indicating numerical problems displayed figure fig similar fig case terms shape expected however domain magnitude decreases shifted factor due discretization model furthermore fig shows maximum size parameter box almost independent also seen case vii onclusion paper proposed analysis robust stability robust state convergence robust performance linear systems uncertain parameters affine quadratic stability certified finding lyapunov function affine parameter order get tractable solution argument extended terms handle discretetime case simulation example figure maximum guarantee aqs method argument seems less conservative quadratic stability slack variable approach future research results extended lyapunov functions quadratic dependence scheduling signal similar case alternatively result extended incremental stability framework lpv systems bounded rates variations eferences shamma analysis design gain scheduled control systems phd thesis massachusetts institute technology mohammadpour scherer control linear parameter varying systems applications springer briat linear systems analysis observation filtering control springer zhou doyle glove robust optimal control prentice hall ebihara peaucelle arzelier approach lmibased robust control springer london megretski rantzer system analysis via integral quadratic constraints ieee trans automatic control vol helmersson stability criterion systems slowly varying parameters proc ifac world congress beijing china jun powers reznick new bound theorem applications polynomials positive polyhedra pure applied algebra vol molchanov liu robust absolute stability nonlinear systems ieee trans circuits systems fundamental theory applications vol scherer robust performance analysis structured linear perturbations bounded ieee trans automatic control vol gahinet apkarian chilali affine lyapunov functions real parametric uncertainty ieee trans automatic control vol chilali gahinet apkarian robust pole placement lmi regions ieee trans automatic control vol mahmoud systems linear stability vand mathematical analysis applications vol oliveira geromel liu hsu lmi characterization structural robust stability case linear algebra applications vol blanchini miani new class universal lyapunov functions control uncertain linear systems ieee trans automatic control vol daafouz bernussou parameter dependent lyapunov functions discrete time systems time varying parametric uncertainties systems control letters vol souza barbosa neto robust filtering linear systems uncertain parameters ieee trans signal processing vol amato mattei pironti gain scheduled control systems depending bounded rate parameters int robust nonlinear control vol iwasaki shibata lpv system analysis via quadratic separator uncertain implicit systems ieee trans automatic control vol peaucelle arzelier henrion gouaisbaut quadratic separation feedback connection uncertain matrix implicit linear transformation automatica vol wei lee stability analysis discrete lpv systems subject parameters proc ifac world congress seoul korea jul oliveira skelton stability tests constrained linear systems perspectives robust control reza moheimani springer london scherer lmi relaxations robust control european control vol veenman scherer robust stability performance analysis based integral quadratic constraints european control vol khalil nonlinear systems prentice hall rawlings mayne model predictive control theory design nob hill publishing fifth scherer weiland linear matrix inequalities lecture notes course dutch institute systems control mar modeling identification linear systems springer oliveira bernussou geromel new discretetime robust stability condition systems control letters vol trofino souza biquadratic stability uncertain linear systems ieee trans automatic control vol scorletti fromion hillerin toward nonlinear tracking rejection using lpv control proc ifac workshop linear parameter varying systems grenoble france ppendix sufficiency let global maximizer assume vertex since global maximizer holds max hand condition imposes convexity implied therefore maximum interval edges max combining leads max concluding maximum obtained edges case satisfied repeating argument implies maximum function vertex condition satisfied
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jun blind nonnegative source separation using biological neural networks cengiz sreyas dmitri center computational biology flatiron institute new york iit madras chennai india nyu medical school new york abstract blind source separation extraction independent sources mixture important problem artificial natural signal processing address special case problem sources mixing matrix known nonnegative example due physical nature sources search solution problem implemented using biologically plausible neural networks specifically consider online setting dataset streamed neural network novelty approach formulate blind nonnegative source separation similarity matching problem derive neural networks similarity matching objective importantly synaptic weights networks updated according biologically plausible local learning rules introduction extraction latent causes sources complex stimuli essential making sense world stimuli could mixtures sounds mixtures odors natural images supervision ground truth causes lacking problem known blind source separation blind source separation problem solved assuming generative model wherein observed stimuli linear combinations independent sources approach known independent component analysis ica jutten herault comon bell sejnowski oja formally stimulus time expressed vector ast unknown mixing matrix represents signals sources time paper assume goal ica infer source signals stimuli whereas many ica algorithms developed signal processing community comon jutten implemented biologically plausible neural networks yet brains solve blind source separation problem effortlessly bronkhorst asari narayan bee micheyl mcdermott mesgarani chang golumbic isomura therefore discovering biologically plausible ica algorithm important problem algorithm implementable biological neural networks must satisfy least following requirements first must operate online streaming setting words input dataset available whole streamed one data vector time corresponding output must computed next data vector arrives second output neurons brain either firing rate synaptic vesicle release rate nonnegative third weights synapses neural network must updated using local learning rules depend activity corresponding postsynaptic neurons given nonnegative nature neuronal output consider special case ica sources assumed nonnegative termed nonnegative independent component analysis nica plumbley course recover sources one use standard ica algorithms rely nonnegativity sources fastica oja hyvarinen oja neural learning rules proposed ica linsker eagleman isomura toyoizumi references within however taking account nonnegativity may lead simpler efficient algorithms plumbley plumbley oja oja plumbley yuan oja zheng ouedraogo existing nica algorithms met biological plausibility requirements terms online setting local learning rules two notable exceptions first plumbley succesfully simulated neural network small dataset yet theoretical analysis given second plumbley plumbley oja proposed nonnegative pca algorithm streaming setting however neural implementation requires nonlocal learning rules algorithm requires prewhitened data see also yet streaming whitening algorithm given propose biologically plausible nica algorithm novelty approach algorithm derived similarity matching principle postulates neural circuits map similar inputs similar outputs pehlevan previous work proposed various objective functions find similarity matching neural representations solved optimization problems biologically plausible neural networks pehlevan pehlevan chklovskii pehlevan chklovskii apply networks nica rest paper organized follows section show blind source separation generalized prewhitening step posed nonnegative similarity matching nsm problem pehlevan chklovskii section using results pehlevan chklovskii show generalized prewhitening step nsm step solved online neural networks local learning rules stacking two networks leads nica network section compare performance algorithm ica nica algorithms various datasets offline nica via nsm section first review plumbley analysis nica reformulate nica nsm problem review plumbley analysis source signals nonnegative source separation problem simplifies solved two straightforward steps noncentered prewhitening orthonormal rotation plumbley noncentered prewhitening transforms whitening matrix hhi hhi angled brackets denote average source distribution identity matrix note mean removed tranformation otherwise one would able use constraint sources nonnegative plumbley assuming sources unit hsi hsi analysis plumbley plumbley assumed mixture channels source channels assumption relaxed shown without loss generality scalar factor multiplies source always absorbed corresponding column mixing matrix mixtures sources prewhitened figure illustration nica algorithm two source channels left linearly transformed mixture middle prewhitening right gives orthogonal transformation sources sources recovered solving nsm problem green red plus signs track two source vectors across mixing prewhitening stages combined effect source mixing prewhitening fas orthonormal rotation see note definition second step nica relies following observation plumbley theorem plumbley suppose sources independent nonnegative wellgrounded prob consider orthonormal transformation permutation matrix probability nonnegative second step look orthonormal nonnegative found plumbley theorem guarantees qfx permutation sources several algorithms developed based observation plumbley plumbley oja oja plumbley yuan oja note sources necessarily mixing matrix must nonnegative therefore nica allows generative models features add also cancel presence shadow image plumbley respect nica general nonnegative matrix factorization nmf lee seung paatero tapper sources mixing matrix required nonnegative nica nsm next reformulate nica nsm problem reformulation allow derive online neural network nica section main departure plumbley analysis work similarity matrices rather covariance matrices finite number samples rather full probability distribution sources first let switch matrix notation data matrices formed concatenating data column vectors notation introduce operation example stimuli dimensional column vector whose elements goal recover unknown make following two assumptions first sources nonnegative decorrelated note general ica nica problems stated independence assumption sources purposes sufficient decorrelated second mixing matrix propose source matrix recovered following two steps also illustrated fig generalized prewhitening transform unit eigenvalues zero eigenvalues whitened otherwise channels correlated prewhitening useful implies according following theorem theorem obeys eigenvalue decomposition diag proof see sufficient first note follows definition holds turn sufficient prove see assume svd decomposition implies diagonal elements gives desired results remark channels correlated except special case whitening used plumbley analysis plumbley requires nsm solve following optimization problem arg min optimization performed nonnegative call nsm cost function pehlevan chklovskii inner products quantify similarities call input output similarity matrices elements hold pairwise similarities input pairwise similarities output vectors respectively cost function preserves input similarity structure much possible nonnegativity constraint variants considered applied math literature name nonnegative symmetric matrix factorization clustering applications kuang make key observation using theorem rewrite arg min since nonnegative matrices permutation matrix solution optimization problem sources successfully recovered uniqueness solutions permutations hard establish sufficient conditions necessary sufficient conditions uniqueness exist verify usually verification donoho stodden laurberg huang review related uniqueness results found huang necessary condition uniqueness given huang states factorization unique permutations row contains least one element equal necessary condition similar plumbley requirement used proving theorem nsm problem solved projected gradient descent max max operation applied elementwise size gradient step algorithms found kuang huang derivation nica neural networks similarity matching objectives analysis previous section revealed nica problem solved two steps generalized prewhitening nonnegative similarity matching derive neural networks steps stack give biologically plausible neural network operates streaming setting departure previous section number output channels reduced number sources prewhitening stage rather later nsm stage assumption simplifies analysis significantly full problem addressed appendix noncentered prewhitening streaming input setting derive neurally plausible online algorithm prewhitening pose generalized prewhitening optimization problem online minimization optimization problem gives algorithm mapped operation biologically plausible neural network generalized prewhitening solves constrained similarity alignment problem max tit centered mixture independent sources matrix constrained tit positive semidefinite solution objective aligns similarity matrices right singular vectors pehlevan chklovskii objective trace diagonalizes value sum eigenvalue pair products since eigenvalues upper bounded objective maximized setting eigenvalues pair top eigenvalues rest zero hence optimal satisfies generalized prewhitening condition pehlevan chklovskii formally theorem modified pehlevan chklovskii suppose eigen decomposition eigenvalues sorted decreasing order optimal svd decomposition form ones top diagonal zeros rest diagonal theorem implies first hence satisfies generalized prewhitening condition second linear mapping constructed using svd decomposition constraint introduced objective function using lagrange multiplier grassmanian matrix min max tit optimization problem pehlevan chklovskii used derive online neural algorithm whereas optimization problem formulated offline setting outputs computed receiving inputs derive biologically plausible algorithm need formulate optimization problem online setting algorithm receives inputs sequentially one time computes output next input arrives therefore optimize data already received respect current output arg min arg max keeping terms depend get following objective arg min arg max limit first three terms dominate last term ignore remaining objective strictly concave convex assume matrix objective unique saddle point wtgh wthg wtgh wthx wthx wtgh wthg wthg wthx interpreted current estimate hence wthg wtgh prewhitening matrix solve gradient wthx wthg wtgh measures time within single time step biologically justified activity dynamics converges faster correlation time input data dynamics proved converge saddle point objective see appendix equation describes dynamics neural network fig wthx represents weights feedforward synaptic connections wthg wtgh represent weights synaptic connections two populations remarkably synaptic weights appear online algorithm despite absence optimization problem formulations furthermore neurons associated principal neurons biological circuit neurons interneurons finally using definition synaptic weight matrices formulate recursive update rules wthx wthg wthg wtgh wthx equations define neural algorithm proceeds two phases stimulus presentation first iterated convergence dynamics neuronal activities second synaptic weights updated according local synapses interneurons hebbian synapses rules biologically synaptic weights updated slower timescale neuronal activity dynamics mixing prewhitening nsm principal hebbian synapses figure biologically plausible network blind source separation prewhitening stage composed two populations linear neurons nsm stage single population rectifying neurons algorithm viewed special case algorithm proposed plumbley plumbley analyzed convergence synaptic weights plumbley stochastic setting linear stability analysis stationary point synaptic weight updates results directly applicable algorithm show synaptic weights algorithm converge stationary state whiten input importantly unlike plumbley proposed algorithm heuristically derived posing solving optimization problem computing optimization problem corresponding neural algorithm eqs almost achieve needed noncentered prewhitening still need find since nsm step need discuss learned along using online algorithm online algorithm following form first neural dynamics stage outputs linear transformation input second synaptic weights hence updated supplement algorithm running estimate let running estimate average stimulus activity alternatively combined single step network receives uncentered stimuli prewhitenes note assignment still done iterating except input rather however synaptic weights still updated using therefore keep recursive estimates means substituting using term requires calculations assuming limit updates small ignore term obtain recursion finally similar argument given keep recursive estimate summarize new stimulus observed algorithm operates two steps first step following neural dynamics runs convergence fixed point dht wthx wthg dgt wtgh convergence proof neural dynamics given appendix also applies besides synaptic weight neuron remembers average activity synapse remembers average incoming activity second step algorithm average activities updated synaptic weight matrices updated recursively wthx wthg wthg wtgh wtgh synaptic updates done new stimulus processed note synaptic update rules local hence biologically plausible wthx online nsm next derive network solves nsm optimization problem online setting pehlevan chklovskii online optimization problem arg min proceeding let rewrite keeping terms depend arg min kyt kht kyt limit last two terms ignored remainder quadratic form minimize using coordinate descent wright fast neurally plausible approach neurons updated performing exact minimization respect convergence max next time step update synaptic weights recursively giving following local learning rules interestingly update rules local identical oja rule oja except learning rate given explicitly terms cumulative activity lateral connections arrival data vector operation complete network algorithm fig follows first dynamics prewhitening network runs convergence output prewhitening network fed nsm network nsm network dynamics runs convergence fixed point synaptic weights updated networks processing next data vector nica stationary state online nsm show solution nica problem stationary synaptic weights state online nsm algorithm stationary state expected updates synaptic weights zero dropped index brackets denote averages source distribution suppose stimuli obey nica generative model observed mixture whitened exact generalized prewhitening matrix described theorem input network time fxt fast claim exists synaptic weight configurations mixed input output network source vector pst permutation matrix synaptic configuration stationary state prove claim constructing synaptic weights permutation matrix first relabel outputs ith output recovers ith source hence becomes identity matrix weights wijy hsi hsj wijy hsi hsk hsi hsi given mixture nsm neural dynamics weights converge unique fixed furthermore weights define stationary state defined assuming fixed learning rate see substitute weights last two equations average source distribution fixed learning rate assumption valid limit changes become small see pehlevan numerical simulations present numerical simulations neural network using various datasets compare results algorithms simulations except fig networks initialized follows prewhitening network whx whg chosen random orthonormal matrices wgh initialized whg definition fact choice guarantees convergence neural dynamics see appendix nsm network initialized random orthonormal matrix set zero learning rates chosen follows prewhitening network generalized learning rate performed grid search find combination best performance performance measures introduced proof net input neuron claimed fixed point wijy wijy plugging fast using one gets net input also output since sources nonnegative fixed point unique globally stable nsm neural dynamics coordinate descent strictly convex cost given nsm network generalized learning rate min performed grid search several values find combination best performance parameter introduces forgetting system pehlevan hypothesized forgetting beneficial setting prewhitening layer output changes time nsm layer adapt comparison purposes also implemented algorithm learning rate form performed grid search find combination best performance nsm network speed simulations implemented procedure plumbley oja iteration checked whether output neuron fired iteration flipped sign feedforward inputs practice flipping occured within first iterations comparison implemented five algorithms first offline algorithm two chosen represent major online algorithm classes offline projected gradient descent simulated projected gradient descent algorithm used variable stepsizes form performed grid search find combination best performance initialized elements matrix drawing gaussian random variable zero mean unit variance rectifying input dataset whitened offline passing projected gradient descent fastica fastica oja hyvarinen oja popular ica algorithm assume nonnegativity sources implemented online version fastica oja using parameters except feedforward weights used learning rate performed grid search find combination best performance fastica requires whitened centered input oja computes decoding matrix maps mixtures back sources ran algorithm whitened centered input recover nonnegative sources applied decoding matrix noncentered whitened input infomax ica bell sejnowski proposed blind source separation algorithm maximizes mutual information inputs outputs namely infomax principle linsker simulated online version due amari chose cubic neural nonlinearities compatible input sources differs fastica implementation nonlinearity also learned online infomax ica computes decoding matrix using centered whitened data recover nonnegative sources applied decoding matrix noncentered inputs finally rescaled sources variance experimented several learning rate parameters finding optimal performance linsker network linsker proposed neural network local learning rules infomax ica simulated algorithm cubic neural nonlinearities preprocessing decoding done infomax ica implementation nonnegative pca nonnegative pca algorithm plumbley oja solves nica task makes explicit use nonnegativity sources use online version given plumbley oja speed simulations implemented procedure plumbley oja iteration checked whether output neuron fired iteration flipped sign feedforward inputs algorithm used learning rate performed grid search find combination best performance nonnegative pca assumes whitened centered input plumbley oja next present results simulations three datasets mixture random uniform sources source samplesq set zero probability sampled uniformly iterval probability dimensions source vectors mixing matrices given appendix source vectors generated run sample original mixed signals see fig inputs fastica nonnegative pca algorithms prewhitened offline case fastica also centered ran nsm network single layer algorithm prewhitening done offline part algorithm whitening done online sources mixtures sample number error sample gradient step sample number sample number error sample number recovered whitened online fastica infomax ica linsker network nonnegative pca nsm nsm two layer network offline projected gradient descent sample gradient step figure performance algorithms presented mixture random uniform sources sample source mixture whitened recovered signals task performed algorithm whitened signal output first layer recovered signal output second layer performance comparison online algorithms presented paper projected gradient descent online fastica infomax ica linsker network nonnegative pca curves show averages simulations error bars shown visual clarity learning rate parameters given appendix quantify performance tested algorithms used permutation matrix chosen minimize learning rate parameters networks optimized grid search using performance metric fig show performances algorithms implemented algorithms perform well better others especially dimensionality input increases offline whitening better online whitening however dimensionality increases online whitening becomes competitive offline whitening fact networks perform better online fastica nonnegative pca whitening done offline also simulated fully offline algorithm taking projected gradient descent steps residual error plateaued fig performance offline algorithm quantifies two important metrics first establishes loss performance due online opposed offline processing second establishes lowest error could achieved nsm method given dataset lowest error necessarily zero due finite size dataset method perfect projected gradient descent may get stuck local minimum also tested whether learned synaptic weights network match theoretical predictions fig show examples learned feedforward recurrent synaptic weights expected theory observed almost perfect match two fig quantify convergence simulated synaptic weights theoretical prediction plotting normalized error metric defined kwt simulation wtheory kwtheory mixture random uniform exponential sources algorithm demix sources sampled different statistical distributions illustrate point generated dataset two uniform three exponential source channels uniform sources sampled exponential sources either zero probability sampled exponential distribution scaled variance channel fig show algorithm succesfully recovers sources test denoising capabilities algorithm created dataset source signals accompanied background noise sources recovered three exponential channels sampled background noises two uniform channels sampled except scaled wyh wyy theory simulation wyh wyy error error sample sample figure theoretical predictions learned synaptic weights match simulations example synaptic weight matrices predicted theory compared results example simulation convergence simulated network synaptic weights theoretical predictions figure inputs whitened offline nsm network run learning rates shaded bars show standard error simulations error sample sample figure performance two layer algorithm presented mixture random uniform exponential sources recovery mixture three exponential two uniform sources recovery three exponential sources corrupted two background noise channels learning rate parameters given appendix variance denoise resulting five dimensional mixture prewhitening layer reduced five input dimensions three nsm layer succesfully recovered sources fig hence prewhitening layer act denoising stage mixture natural scenes next consider recovering images mixtures fig image treated one source four image patches size pixels chosen set images natural scenes previously used hoyer plumbley oja preprocessing plumbley oja images downsampled factor obtain patches pixel intensities shifted minimum zero pixel intensities scaled unit variance hence dataset sources corresponding image patches total samples samples presented algorithm times randomly permuted order presentation mixing matrix generated randomly given appendix fig show performances algorithms implemented task see algorithms compared fastica nonnegative source images mixtures recovered images online fastica nonnegative pca nsm two layer network sample figure performance algorithm presented mixture natural images sample source mixture recovered images performed algorithm performance comparison online algorithms online fastica nonnegative pca shaded bars show standard error simulations learning rate parameters listed appendix pca perform much better discussion paper presented new neural algorithm blind nonnegative source separation started assuming nonnegative ica generative model plumbley inputs linear mixtures independent nonnegative sources showed sources recovered inputs two sequential steps generalized whitening nsm fact argument requires sources uncorrelated necessarily independent two steps performed online neural networks local learning rules pehlevan chklovskii stacking two networks yields neural network blind nonnegative source separation fig numerical simulations show neural network algorithm performs well network derived optimization principles biologically realistic features given meaning network layer performs different optimization lateral connections create competition principal neurons forcing differentiate outputs interneurons clamp activity dimensions principal neurons pehlevan chklovskii rectifying neural nonlinearity related nonnegativity sources synaptic plasticity malenka bear implemented local hebbian learning rules achieves online learning hebbian learning famously observed neural circuits bliss bliss network also makes heavy use learning interpreted potentiation inhibitory postsynaptic potentials experiments show potentiation arise pairing action potentials inhibitory neurons subthreshold depolarization postsynaptic pyramidal neurons komatsu maffei however plasticity inhibitory synapses hebbian require correlation postsynaptic activity kullmann improved biological realism network respond continuous stimulus stream continuous simultaneous changes outputs synaptic weights presumably requires neural time scales faster synaptic time scales slower changes stimuli explore possibility simulated datasets limited number neural activity updates shown found updates per neuron sufficient successful recovery sources without significant loss performance neural time scale take sufficiently fast given example temporal autocorrelation time scale natural image sequences david bull interesting compare architecture present multilayer neural networks deep learning approaches lecun data presentation network performs recurrent dynamics produce output deep networks feedforward architecture first layer network multiple neuron types principal interneurons principal neurons project next layer deep learning neurons layer project next layer network operates local learning rules deep learning uses backpropagation local derived architecture dynamics learning rules network principled cost function deep learning architecture dynamics neural network designed hand learning rule derived cost function finally building neural algorithm started generative model inputs inferred algorithmic steps recover latent sources algorithmic steps guided deciding networks stack deep learning generative model assumed network architecture design art believe starting generative model might lead systematic way network design fact question generative model appropriate deep networks already asked patel acknowledgments thank andrea giovannucci eftychios pnevmatikakis anirvan sengupta sebastian seung useful discussions grateful iarpa microns program support convergence gradient dynamics prove neural dynamics converges saddle point objective function assume whg first note optimum also fixed point since neural dynamics linear fixed point globally convergent eigenvalues matrix wgh negative real parts eigenvalue equation wgh implies wgh using relations solve eigenvalues two cases implies wgh whg wgh since wgh dimensional one degenerate eigenvalues substituting first equation implies whg wgh hence eigenvector whg wgh eigenvalue two corresponding eigenvectors solved uniquely first equation hence dqdegenerate eigenvalues pairs conjugate eigenvalues one pair eigenvaleue whg wgh since real positive assume whg definition whg wgh real parts negative hence neural dynamics globally convergent modified objective function neural network generalized prewhitening deriving online neural algorithm assumed number output channels reduced number sources prewhitening stage however offline analysis need reduction one could keep generalized prewhitening provide online neural algorithm allows first point prewhitening algorithm given main text adequate appendix proved neural dynamics described converges saddle point objective function proof assumes whg however assumption breaks network learns perfectly prewhitened rank perfectly prewhitening network would whg would also simulated network observed condition number whg wgh increased neural dynamics took longer time converge even though algorithm still functioning well practical purposes present modification fully resolves problem propose modified offline objective function pehlevan chklovskii corresponding neural network consider following min max centered mixture independent sources matrix matrix positive parameter notice additional term compared less lowest eigenvalue optimal linear transform satisfies generalized prewhitening condition pehlevan chklovskii precisely theorem modified pehlevan chklovskii suppose eigen decomposition eigenvalues sorted order magnitude less lowest eigenvalue optimal svd decomposition form ones top diagonals zeros rest using cost function derive neural algorithm suffer described convergence issues even hand need choose parameter need know spectral properties derive online algorithm repeat steps taken arg min arg max limit first four terms dominate last term ignore remaining objective strictly concave strictly convex note convex strictly convex objective unique saddle point even wtgh wthg wtgh wthx matrices defined identity matrix solve gradient wthx wthg wtgh time measured within single time step dynamics proved converge saddle point modifying proof appendix synaptic weight updates finally network modified also compute following steps fixed point globally convergent eigenvalues matrix wgh negative real parts one show eigenvalues eigenvalues positive eigenvalue whg wgh one gets pair eigenvalues negative real parts mixing matrices numerical simulations random source dataset mixing matrix list mixing matrices cases purposes however available authors upon request natural scene dataset mixing matrix learning rate parameters numerical simulations figs following parameters found best performing result grid search fastica npca nsm activity nsm time images images offline infomax ica linsker algorithm references amari cichocki yang new learning algorithm blind signal separation advances neural information processing systems asari pearlmutter zador sparse representations cocktail party problem journal neuroscience bee micheyl cocktail party problem solved animal behaviorists study journal comparative psychology bell sejnowski approach blind separation blind deconvolution neural computation bliss potentiation synaptic transmission dentate area unanaesthetized rabbit following stimulation perforant path journal physiology bliss potentiation synaptic transmission dentate area anaesthetized rabbit following stimulation perforant path journal physiology bronkhorst cocktail party phenomenon review research speech intelligibility conditions acta acustica united acustica bull communicating pictures course image video coding academic press comon independent component analysis new concept processing signal comon jutten handbook blind source separation independent component analysis applications academic press david vinje gallant natural stimulus statistics alter receptive field structure neurons journal neuroscience donoho stodden matrix factorization give correct decomposition parts advances neural information processing systems page none eagleman coenen mitsner bartol bell sejnowski cerebellar glomeruli limited extracellular calcium implement sparse encoding strategy proceedings annual joint symposium neural computation golumbic ding bickel lakatos schevon mckhann goodman emerson mehta simon mechanisms underlying selective neuronal tracking attended speech cocktail party neuron pehlevan chklovskii network online sparse dictionary learning derived symmetric matrix factorization asilomar conference signals systems computers pages ieee huang sidiropoulos swami matrix factorization revisited uniqueness algorithm symmetric decomposition signal processing ieee transactions hyvarinen fast robust algorithms independent component analysis ieee transactions neural networks hoyer emergence features decomposition natural images independent feature subspaces neural computation oja fast algorithm independent component analysis neural computation oja independent component analysis general nonlinear learning rules signal processing oja independent component analysis algorithms applications neural networks isomura kotani jimbo cultured cortical neurons perform blind source separation according principle plos comput biol isomura toyoizumi local learning rule independent component analysis scientific reports jutten herault blind separation sources part adaptive algorithm based neuromimetic architecture signal processing komatsu potentiation inhibitory synaptic transmission rat visual cortex journal neuroscience kuang park ding symmetric nonnegative matrix factorization graph clustering sdm volume pages siam kuang yun park symnmf nonnegative approximation similarity matrix graph clustering journal global optimization kullmann moreau bakiri nicholson plasticity inhibition neuron laurberg christensen plumbley hansen jensen theorems positive data uniqueness nmf computational intelligence neuroscience lecun bengio hinton deep learning nature lee seung learning parts objects matrix factorization nature liang risteski recovery guarantee matrix factorization via alternating updates arxiv preprint linsker perceptual network computer linsker local learning rule enables information maximization arbitrary input distributions neural computation maffei nataraj nelson turrigiano potentiation cortical inhibition visual deprivation nature malenka bear ltp ltd embarrassment riches neuron mcdermott cocktail party problem current biology mesgarani chang selective cortical representation attended speaker speech perception nature narayan best ozmeral mcclaine dent sen cortical interference effects cocktail party problem nature neuroscience oja simplified neuron model principal component analyzer math biol oja plumbley blind separation positive sources globally convergent gradient search neural computation ouedraogo souloumiac jutten independent component analysis algorithm based givens rotations newton optimization latent variable analysis signal separation pages springer paatero tapper positive matrix factorization factor model optimal utilization error estimates data values environmetrics patel nguyen baraniuk probabilistic framework deep learning advances neural information processing systems pages pehlevan chklovskii network derived online matrix factorization cluster discover sparse features asilomar conference signals systems computers pages ieee pehlevan chklovskii normative theory adaptive dimensionality reduction neural networks advances neural information processing systems pages pehlevan chklovskii optimization theory networks pca whitening annual allerton conference communication control computing allerton pages ieee pehlevan chklovskii neural network linear subspace learning derivation multidimensional scaling streaming data neural computation plumbley subspace network determines output dimension tech plumbley information processing negative feedback neural networks neural plumbley conditions nonnegative independent component analysis signal processing letters ieee plumbley adaptive lateral inhibition ica proceedings international conference independent component analysis blind signal separation pages plumbley algorithms nonnegative independent component analysis neural networks ieee transactions plumbley oja nonnegative pca algorithm independent component analysis neural networks ieee transactions wright coordinate descent algorithms mathematical programming yuan oja fastica algorithm independent component analysis independent component analysis blind signal separation pages springer zheng huang sun lyu lok nonnegative independent component analysis based minimizing mutual information technique neurocomputing
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delay performance miso wireless communications jul arnau member ieee marios kountouris senior member ieee abstract low latency communications urllc currently attracting significant attention due emergence applications communication urllc entail fundamental paradigm shift system design towards holistic designs guaranteed reliable latency deep understanding delay performance wireless networks essential efficient urllc systems paper investigate network layer performance miso systems statistical delay constraints provide expressions miso service process derive probabilistic delay bounds using tools stochastic network calculus particular analyze transmit beamforming perfect imperfect channel knowledge compare orthogonal codes antenna selection effect transmit power number antennas finite blocklength channel coding delay distribution also investigated higher layer performance results reveal key insights miso channels provide useful guidelines design communication systems guarantee stringent urllc latency requirements index terms urllc systems mimo diversity stochastic network calculus finite blocklength channel coding queueing analysis ntroduction data traffic growing tremendously last decade fueled ubiquity smart mobile devices applications order handle everincreasing traffic load existing wireless networks typically designed planned authors mathematical algorithmic sciences lab france research center huawei technologies france sasu quai point jour france email focus improving spectral efficiency increasing coverage latency requirements different applications mostly reliability low latency mainstream wireless networks due focus humancentric communications content reliability levels order however plethora socially useful applications new uses wireless communication currently envisioned areas industrial control smart cities augmented virtual reality automated driving flying robotics telemedicine algorithmic trading tactile internet response new releases mobile cellular networks mainly new radio beyond envisaged support low latency communications urllc scenarios strict requirements terms latency ranging milliseconds latency depending use cases reliability higher another new feature support communications mtc massive number connected devices transmit reliably relatively low volume payload information theory communication engineering instrumental boosting spectral efficiency approaching capacity limits nevertheless urllc communication pose significant theoretical practical challenges requiring departure system design towards holistic view network architecture control data guaranteed reliable latency applying information theory design low latency networks challenge information theory mostly focuses asymptotic limits achieved arbitrarily small probability error using long codewords hence arbitrarily large coding delays despite recent development block error rates finite blocklength codes work needed better understand fundamental tradeoffs delay reliability throughput including coding delays queueing delays addition highly variable nature network traffic together associated overhead metadata incorporated conventional communication theoretic framework reliable communication problem dating back shannon landmark paper techniques usually employed increase reliability combating exploiting channel variations several schemes developed including error correction codes use multiple antennas diversity physical layer well automatic arq opportunistic scheduling erasure coding higher layers among mimo systems received great interest due potential combat fading increase spectral efficiency reduce interference mimo techniques used beam steering diversity spatial multiplexing interference cancellation techniques increase reliability combating exploiting channel variations beam steering techniques increase received signal quality focusing desired energy attenuating undesired interference spatial multiplexing increases data rate transmitting independent data symbols across antennas work focus diversity beam steering techniques relevant reliable communications particular consider maximum ratio transmission mrt transmit beamforming technique maximizes received signal realizes diversity exploiting channel state information csi transmitter several attempts addressing latency considerations physical layer including error exponents capacity outage capacity tradeoff curves finite blocklength channel coding networking delay key performance measure queueing theory instrumental providing exact solutions backlog delays networks however queueing network analysis largely restricted networks interacting coupled queues small topologies poisson arrivals classical queueing models typically allow analysis average delay failing characterize delay quantiles delay distributions cardinal importance applications recent efforts combine queueing communication theory stochastic network calculus timely throughput effective bandwidth effective capacity name take different approach compute performance bounds wide range stochastic processes approaches promise significant performance gains terms latency reliability throughput crisp insights design low latency communication systems work employ stochastic network calculus probabilistic extension deterministic network calculus allows stochastic bounds network performance metrics maximum delay broad classes arrival scheduling service processes despite extended literature mimo techniques physical layer attempts made characterize upper layer performance techniques taking account queueing effects service process adaptive mimo system poisson arrivals characterized bounds delay violation probability derived mimo multiple access bursty traffic provides asymptotic analysis tradeoff mimo systems bursty information using large deviations analyzes queueing performance scheduling multiuser mimo systems bounds tail delay mimo communication systems derived using effective capacity framework nevertheless approximations valid large delays constant bit rate processes using markov chains reproduce state fading channels performance mimo spatial multiplexing analyzed using stochastic network calculus nevertheless none works considered delay performance mimo schemes using stochastic network calculus wireless fading channels work study upper layer delay performance multiple input single output miso diversity communication presence statistical delay constraints consider mrt transmit beamforming physical layer derive probabilistic delay bounds using tools stochastic network calculus provide characterization cumulative service process miso beamforming channels perfect imperfect csi analysis use min network calculus methodology wireless network delay analysis impact transmit antennas ratio snr imperfect csi delay distribution miso mrt systems characterized show mathematical framework applied statistical characterization various mimo service processes including mimo orthogonal block coding ostbc antenna selection fading channels allows compare delay performance transmit beamforming alternative techniques rely low rate csi transmit antenna selection csi ostbc interestingly miso mrt shown reduce delay violation probability compared transmissions even imperfect csi derived delay bounds enable assess robustness miso mrt delay performance respect channel imperfections results also show operating parameters preferable mrt terms delay violation probability addition provide asymptotic statistical characterization service process snr regime large number antennas finally extending miso systems study effect finite blocklength channel coding queueing delay performance results quantify performance loss due finite blocklength characterize tradeoff data rate error probability respect delay performance results provide useful insights guidelines design wireless systems satisfy guarantee stringent urllc latency requirements rest paper organized follows section provide system model section iii brief background min network calculus presented section delay performance analysis miso diversity systems derived section provides delay performance asymptotic regimes section shows effect finite blocklength delay performance numerical results presented section vii followed conclusions section viii ystem model consider data transmission vector communication channel time divided time slots duration model slot source generates data bits stores queue transmitter source transmit antennas sends queued data bits receiver rayleigh fading channel assume model channel remains constant one slot varies independently slot slot slot contains symbols denotes complex data symbols metadata headers training estimation acknowledgments signal model received downlink signal slot miso wireless channel given snr channel transmitter receiver slot complex gaussian distributed transmitted vector denoted additive background noise may also include gaussian interference neighboring systems consider one prominent diversity technique namely transmit beamforming refers sending linearly weighted versions signal antenna transmitted signal written data signal slot power beamforming vector note since noise assumed unit power snr represents average received snr whereas instantaneous snr slot given transmission mode natural signaling strategy miso channel maximize snr specific channel achieved sending information direction channel vector information sent orthogonal direction nulled channel anyway thus consider maximum ratio transmission mrt equivalent since beamforming along dominant eigenmode vector channel assuming transmitter receiver perfect csi mrt beamforming vector given khi case instantaneous snr snrkhi gamma distributed shape parameter scale parameter snr gamma snr transmitter fully know actual channel vector imperfect csi model channel knowledge mrt performed based particularizing miso case instantaneous snr gamma distributed shape parameter scale parameter gamma snr channel estimate additive error model consistent duplex tdd operation uplink downlink transmissions take place frequency different time instants assuming fall within coherence interval channel channel reciprocity used estimate downlink channel uplink pilot signals model also applies frequencydivision duplex fdd operation analog feedback account effect csi error miso beamforming reduces achieved snr snr loss transmitting exactly direction actual channel explain next subsection could another penalty rate selection process shown mrt transmission scheme maximizes capacity service rate sum power constraint gaussian input signaling perfect csi necessarily true arbitrary inputs transmission scheme may depend delay constraints data transmission codeword length symbols corresponding channel uses rate bits per symbol transmitted slot transmitter selects rate adapted consider following two cases asymptotically large blocklength blocklength large enough errors occur achievable rate equal shannon capacity channel finite blocklength finite blocklength transmission error occur probability maximum coding rate lower shannon rate tight non asymptotic upper lower bounds maximum coding rate given furthermore awgn channels asymptotic approximation established shown accurate packet sizes small coding rate transmit information bits using coded packets spanning channel uses given inverse gaussian function channel dispersion given using approximation coding rate packet error probability related approximation valid miso systems knowledge fading coefficients vector channel snr realization slot makes miso channel behaving equivalently awgn channel snr remark far throughout section assumed perfect knowledge snr realization transmitter adapt rate errors thus account channel estimation error snr penalty described sec case imperfect rate selection miso systems goes beyond scope work analyzed using techniques recently developed case channel estimation errors approximated gaussian variations snr gaussian variations capacity due finite blocklength transformed gaussian errors snr queuing model analysis queuing systems consider stochastic model widely used stochastic network calculus methodology stochastic network calculus considers queuing systems networks systems stochastic arrival departure service processes ones description topic interested reader may refer arrival process introduced sec models number bits arrive queue discrete time instant successful transmissions service process equal nri asymptotically large blocklength nri finite blocklength case transmission errors service considered zero data removed queue finite blocklength channel coding affects reliability physical layer turn causes additional delay data needs buffered successfully transmitted departure process describes number bits arrive successfully destination depends service process number bits waiting queue note acknowledgments feedback messages assumed instantaneous define cumulative arrival service departure processes lossless queuing systems delay time number slots takes information bit arriving time received destination defined inf delay violation probability given using dynamic server property min convolution operator defined inf delay characterized cumulative arrival service processes far described bit domain convenient analysis wireless fading channels follow analyze processes exponential snr domain iii tochastic etwork alculus snr omain remarkable feature stochastic network calculus snr domain allows obtain bounds delay violation probability based simple statistical characterizations arrival service processes terms mellin transforms briefly review result section let start converting cumulative processes bit domain exponential function corresponding processes snr domain denoted calligraphic letters definitions upper bound delay violation probability computed means mellin transforms inf kernel defined lim denotes mellin transform nonnegative random variable expectation exists restrict derivations work recall continuous probability density function pdf exists alternatively mellin transform arrival service processes assuming stationary independent increments mellin transforms become independent time instance follows eai defined arrival process snr domain consider traffic class arrivals whose moment generating function bit domain bounded log esa restricting case independent service process start rewriting log log since different independent identically distributed express mellin transform cumulative service delay bound plugging following kernel finally rewritten stability condition delay bound thus reduces inf elay arge locklength xact nalysis section derive exact expressions kernel miso diversity schemes blocklength infinitely large start providing general result mellin transform service process instantaneous snr gamma distributed obtaining kernel mrt beamforming perfect imperfect csi particularization result shown apply byproduct obtaining performance diversity techniques including miso ostbc antenna selection mimo consider instantaneous snr gamma distributed random variable gamma shape parameter scale parameter pdf complete gamma function dropped subindex since snrs independent ergodic first derive mellin transform notation convenience remainder assume log however sec vii give relevant values parameter order obtain meaningful numerical results theorem mellin transform gamma given tricomi confluent hypergeometric function also called confluent hypergeometric function second kind denoted proof see appendix miso mrt mellin transform derived applies directly service process miso mrt transmission using expression together transform arrival process obtain kernel consequently bound delay violation probability follows mmrt inf mmrt inf although implemented standard software mathematical calculations provide alternative expression mellin transform terms simpler upper incomplete gamma function theorem mellin transform theorem given upper incomplete gamma function proof see appendix remark siso case letting snr obtain snr expression reported expressions allows obtain bounds delay violation probability different system parameters without resorting monte carlo simulations however due complexity kernel function solution minimum found must resort numerical methods asymptotic cases simpler expressions mellin transform make process easier show later section miso diversity techniques far considered transmitter performs mrt based perfect imperfect csi section means comparison study two alternative diversity techniques namely ostbc transmit antenna selection rely csi respectively ostbc orthogonal block coding successful transmit diversity technique achieve full diversity without csi transmitter need joint decoding multiple symbols characterized number independent symbols transmitted time slots code rate transmitter uses ostbc transmit antennas code parameter receiver performs mrc antennas equivalent snr snr gamma distributed shape parameter scale parameter denotes mimo channel matrix complex gaussian entries particularizing case miso ostbc following result corollary mellin transform service process miso system employing ostbc given mostbc snr antenna selection antenna selection feedback diversity technique transmitter receiver select subset antennas used conjunction diversity techniques improve performance mimo expense low amount feedback consider transmit antenna selection tas transmitter selects transmit antenna one maximizes instantaneous snr amount csi required fed back transmitter bits index best antenna dxe denotes smallest integer larger instantaneous snr expressed largest channel gain max since exponentially distributed unit mean pdf theorem mellin transform miso system employing tas given tas proof see appendix applications theorem section briefly point towards possible applications theorem siso case fading theorem could easily used analyze siso case fading distribution includes special cases rayleigh fading ricean distribution ricean factor envelope received signal distributed instantaneous snr gamma distributed shape parameter rate parameter thus mellin transform simply mimo consider receiver equipped receive antennas perform transmitter receiver ends order maximize snr receiver transmit weighting vector selected eigenvector wishart matrix corresponds largest eigenvalue corollary mellin transform service process case mimo mrt rayleigh fading given mimo coefficients obtained perfect csi case tabulated values table result direct consequence note since largest eigenvalue complex wishart matrix equivalently maximum singular value bounded min gamma simple upper lower bounds mellin transform mimo obtained particularizing theorem elay arge locklength symptotic nalysis previous section provided analytical expressions mellin transform service process kernel various diversity techniques exact results mainly given terms special functions alternating series explore delay performance miso mrt derive section simplified expressions various asymptotic regimes snr large additionally obtain general result gaussian distributed service process show miso mrt service process converges distribution grows large high snr regime study latency constraints affect miso performance high snr assume implies also large true long snr see matter fact frequently practice corollary high snr regime mellin transform service process scales mhg log denotes digamma function sec proof three branches obtained direct application asymptotic properties function listed sec first branch also derived considering approximated service process log gives mhg low snr regime low snr service process approximated following result obtained corollary low snr regime mellin transform service process approximately given mlg proof low snr use first order taylor series expansion log case service process approximated gives consequence mlg using moment generating function mgf gamma random variable large antenna regime distribution mutual information rayleigh fading mimo system generally rather complicated reason approximations used literature example large antenna regime using central limit theorem clt shown distribution mutual information converges gaussian distribution see instance references therein using similar arguments obtain simpler expressions mellin transform service process general obtain results form convergence distribution variance term measure convergence speed normally means large accurate approximation distribution given mean variance terms obtained closed form note brevity throughout section use natural logarithm thus rates nats thanks clt arguments gaussian approximation service process arrive following result theorem mellin transform service process rate following gaussian distribution mean variance given masg proof service given terms log bit domain thus eii result follows immediately solving masg theorem miso mrt case number antennas grows large lim masg proof mutual information written log rewriting applying jensen inequality masg log log last equality follows fact lim shown bound asymptotically tight using cam theorem continuous mapping theorem chebyshev inequality proof standard omitted sake brevity furthermore asymptotic convergence obtained without resorting gaussian approximation showing convergence distribution implies convergence let sequence positive random variables converges distribution positive random variable lim myi lebesgue dominated convergence theorem lim interestingly observe large mas related sog called channel hardening effect channel behaves equivalently awgn channel snr low snr regime number transmit antennas affects linearly service process high snr mellin transform service process grows superlinearly approximation allows simplify delay violation probability expression however relevance applicability goes beyond allows analyzing delay violation probability system whose service rate approximated gaussian random variable additionally provides simple expressions effective capacity show next effective capacity effective capacity defined maximum constant arrival rate system support given qos requirement byproduct delay analysis using stochastic network calculus obtain expressions effective capacity noticing log example taking normalized logarithm recover effective capacity results however assess effect multiple antennas performance simpler expressions would beneficial thus focus gaussian approximation sec obtain ras log var log expected effective capacity converges ergodic capacity absence delay constraints general penalty achievable rate proportional variance instantaneous rate log implies number antennas tends infinity penalty vanishes variance rate tends zero effective capacity decay increases elay nalysis inite locklength investigate effect finite blocklength service process delay performance explained sec finite blocklength always probability error rate loss compared shannon capacity case transmission errors offered service zero therefore service process modeled coding rate approximated using bernoulli random variable one case successful transmission probability zero otherwise work assume independence snr using lower bounding achievable rate zero low snr values mellin transform given mfb max given max point max major difficulty deriving fact channel dispersion depends taylor series expansion series expansion exponential function used results easily extended miso case however expression would even involved due gamma distributed channel gains providing little insight numerically evaluate integral required focus simpler asymptotic expression high snr high snr regime high snr channel dispersion approximated mellin transform approximately max last equality followed procedure used obtain theorem see appendix note recover mellin transform service process infinite blocklength approximation requires widely used standard special functions easier evaluate numerically section vii show accuracy satisfactory even moderate values snr vii umerical esults section provide numerical evaluation performance miso communication systems based analysis unless otherwise stated duration slot set overhead disregarded blocklength assumed consequently log reincorporate parameter equations start validating analysis monte carlo simulations figure compare delay violation probability bound kbps snr corroborate delay bounds monte carlo figure delay violation probability associated bounds function target dealy kbps snr bounds follow trend original curve point maximum difference seems figure plots violation bound miso mrt function target delay kbps snr plot left shows effect varying number antennas accuracy csi observe strong decrease delay violation probability increasing number antennas perfect csi probability exceeding delay roughly decreases three orders magnitude adding extra antenna hand plot right depicts difference assuming finite infinite blocklength similar see difference remarkable shannon model substantially overestimates performance system figure compare delay performance miso mrt ostbc tas see mrt generally performs better quality csi good certain value tas ostbc outperform mrt values change takes place seem dependent number antennas obtain results finite blocklength must set error probability experiment used parameters set inspiration figure illustrates importance choosing wisely depending snr also number antennas figure delay violation probability bound function target delay kbps snr curves labeled obtained using finite blocklength expressions ostbc tas figure delay violation probability bound function target delay different diversity techniques kbps snr figure investigate effect adding antennas compare increasing power target delay see going three four antennas seems slightly less impact doubling power however case anymore extra power decrease violation probability one order magnitude adding one antenna decreases two orders magnitude explained section important simple expressions kernel figure bound probability exceeding delay function block error rate finite blocklength analysis kbps circles mark minimum curve figure bound probability exceeding delay function number antennas asymptotically large blocklength kbps exact gaussian approx exact gaussian approx figure mellin transform left kernel right function kbps exact gaussian approx kbps exact gaussian approx kbps figure mellin transform left kernel right function possible figure figure illustrate accuracy gaussian approximation expected error large former negligible latter justifies use much simpler expression whenever relatively large figure left test accuracy high low snr approximations derived shannon model section section see high snr approximation becomes asymptotically tight snr increases remarkably low snr approximation reasonably accurate snr values makes low snr approximation particularly interesting given simplicity see similarly figure right show accuracy high snr approximation derived finite blocklength model high snr approx exact low snr approx hsnr approx exact figure bound probability exceeding delay asymptotically large blocklength left finite blocklength right analysis kbps infinite blocklength finite blocklength blocklength figure bound probability exceeding delay function blocklength kbps circles mark minimum curve section finally show effect varying blockelength assume constant overhead symbols time slot total symbols transmitted duration time slot see figure delay performance heavily depends blocklength chosen optimum value changes number antennas viii onclusions work characterized delay performance miso diversity communications statistical delay constraints using stochastic networks calculus derived statistical characterization service process fading channels provided probabilistic delay bounds showed number transmit antennas transmit snr may affect delay performance also investigated impact imperfect csi transmitter finite blocklength channel coding delay performance miso transmit beamforming miso mrt shown reduce delay violation probability compared transmissions even imperfect csi nevertheless channel imperfections increase ostbc antenna selection perform better mrt terms delay violation probability future work could consider effect imperfect csi receiver limited feedback fdd mimo systems extensions framework may include analysis mimo spatial multiplexing mimo channels interference multiuser mimo systems ppendix roof heorem recall gamma distributed pdf follows definition tricomi confluent hypergeometric function ppendix roof heorem equation rewritten applying change variables since positive integer use binomial theorem obtain alternatively ppendix roof heorem suppose independent continuous variates cdf pdf pdf order statistic given therefore pdf given since exp case tas pdf mtas follows applying binomial theorem eferences ephremides hajek information theory communication networks unconsummated union ieee trans inf theory vol polyanskiy poor verdu channel coding rate finite blocklength regime ieee trans inf theory vol may shannon mathematical theory communication bell system technical journal vol july gallager information theory reliable communication new york usa john wiley sons hanly tse multiaccess fading capacities ieee trans inf theory vol ozarow shamai wyner information theoretic considerations cellular mobile radio ieee trans veh vol may gamal mammen prabhakar shah wireless networks proc ieee infocom mar chang performance guarantees communication networks jiang liu stochastic network calculus london london fidler rizk guide stochastic network calculus ieee commun surveys vol first quarter hou borkar kumar theory qos wireless proc ieee infocom apr negi effective capacity wireless link model support quality service ieee trans wireless vol july cruz calculus network delay part network elements isolation ieee trans inf theory vol zhou zhang niu yang queuing analysis mimo systems adaptive modulation coding proc ieee inter conf commun icc may kittipiyakul javidi optimal operating point mimo multiple access channel bursty traffic ieee trans wireless vol kittipiyakul elia javidi analysis communications bursty information ieee trans inf theory vol chen lau large deviation delay analysis mimo systems feedback ieee trans signal vol gursoy mimo wireless communications statistical queueing constraints ieee trans inf theory vol matthaiou alexandropoulos ngo larsson analytic framework effective rate miso fading channels ieee trans vol jun sun jiang zhang unified framework effective rate analysis wireless communication systems miso fading channels ieee trans vol apr mahmood rizk jiang delay spatial multiplexing mimo wireless channel proc ieee inter conf commun icc june mahmood vehkapera jiang delay constrained throughput analysis correlated mimo wireless channel proc inter conf comp commun netw icccn july liebeherr burchard min network calculus fading channels proc ieee infocom april schiessl gross delay analysis wireless fading channels finite blocklength channel coding proc acm inter conf anal simul wirel mob systems mswim maximum ratio transmission ieee trans vol chen tellambura performance analysis maximum ratio transmission imperfect channel estimation ieee commun vol apr caire jindal kobayashi ravindran multiuser mimo achievable rates downlink training channel state feedback ieee trans inf theory vol jun yang durisi koch polyanskiy fading channels finite blocklength ieee trans inf theory vol july hayashi information spectrum approach coding rate channel coding ieee trans inf theory vol schiessl skoglund gross analysis wireless communications finite blocklength imperfect channel knowledge online available http jiang emstad analysis stochastic service guarantees communication networks server model meer bhatti eds berlin heidelberg springer fidler probabilistic network calculus moment generating functions proc ieee quality service june network calculus approach probabilistic quality service analysis fading channels proc ieee global commun conf globecom petreska gross power minimization industrial wireless networks statistical delay constraints proc international teletraffic congress itc abramowitz stegun handbook mathematical functions formulas graphs mathematical tables new york dover paulraj nabar gore introduction wireless communications new york usa cambridge university press dighe mallik jamuar analysis diversity rayleigh fading ieee trans vol apr hochwald marzetta tarokh channel hardening implications rate feedback scheduling ieee trans inf theory vol mckay smith suraweera collings mutual information distribution spatial multiplexing exact variance outage approximation ieee trans inf theory vol july david nagaraja order statistics hoboken usa john wiley sons
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actors without borders amnesty imprisoned state elias castegren tobias wrigstad uppsala university sweden concurrent systems form synchronisation typically needed achieve freedom important correctness safety systems messages exchanged concurrently executed sequentially receiving actor relying isolation actor access state without fear internal behavior actor reasoned sequentially however actor isolation sometimes strong express useful patterns example letting iterator alias internal structure collection allows efficient implementation access requires going interface collection full isolation order maintain sequential reasoning iterator must made part collection bloats interface collection means client must access whole order use iterator paper propose programming language construct enables relaxation isolation without sacrificing sequential reasoning formalise mechanism simple lambda calculus actors passive objects show actor may leak parts internal state ensuring interaction data still synchronised introduction synchronisation key aspect concurrent programs different concurrency models handle synchronisation differently pessimistic models like locks actor model serialise computation within certain encapsulated units allowing sequential reasoning internal behavior case actor model brevity including also active objects carry state actor traditionally reference actor internal state accessible outside operations inside subject sequential reasoning lost holds true operations aggregate object behind lock subobject leaked becomes accessible appropriate lock held previous work designed kappa type system boundary unit encapsulation statically identified entire encapsulated unit wrapped inside synchronisation mechanism lock asynchronous actor interface consequently operations inside boundary guaranteed free important goal work facilitating reuse concurrent programming internal objects oblivious freedom guaranteed building blocks reused without change regardless external synchronisation extended abstract explores two extensions system explain context actor model although equally applicable system using locks rather rejecting programs actors leak internal objects allow actor bestow synchronisation mechanism upon exposed objects allows multiple objects effectively construct actor interface exposing internal operations externally makes concurrency allow external control possible interleaving operations introduce atomic block groups together following section motivates extensions vasconcelos haller eds workshop programming language approaches software places eptcs castegren wrigstad class node var next node var elem getters setters omitted actor list var first node def getfirst node return def get int var current current return class iterator var current node def init first node void first def getnext val elem return elem def hasnext bool return null actor list def getiterator iterator val iter new iterator return iter figure list implemented actor iterator list breaking isolation motivating example motivate breaking isolation context actor language actors serving units encapsulation encapsulating zero passive objects figure shows kappa program linked list style actor asynchronous external interface simplicity allow asynchronous calls return values omit details accomplished using futures promises passing continuations clients interact list example sending message get specified index implementation time get called corresponding element calculated head list giving linear time complexity access iterating elements list quadratic time complexity allow efficient element access list provide iterator holds pointer current node figure allows access current element linear iteration also breaks encapsulation providing direct access nodes elements without going list interface list operations subject middle ground providing linear time iteration without implemented moving iterator logic list actor calls getnext hasnext synchronised message queue actor requires advanced scheme map different clients different concurrent iterators clutters list interface creates unnecessary coupling list iterator complicates support several kinds iterators another issue concurrent programs interleaving interaction actor makes hard reason operations built several smaller operations example client might want access two adjacent nodes list combine elements somehow sending two get messages nothing prevents messages processed list actor first one possibly removing changing one values actors without borders amnesty imprisoned state actor list def getiterator iterator val iter new iterator return bestow iter val iter list getiterator iter hasnext val elem iter getnext figure list actor returning bestowed iterator code client using unless list actor explicitly provides operation getting adjacent values way client safely express operation bestowing grouping activity encapsulating state behind synchronisation mechanism allows reasoning sequentially operations state however since kappa lets identify encapsulation boundary data structure possible bestow objects leaked across boundary synchronisation wrapper statically means changing type returned reference reflect operations may block dynamically means identifying leaked object shall synchronise clarity explicate pattern bestow operation case actors actor performs bestow reference creates wrapper around makes appear like actor interface asynchronous operations bestowed reference relayed actor one actually performing operation leaked enclosure protected lock wrapper would instead acquire release around operation figure shows minimal changes needed code figure well code client using iterator change list getiterator returns bestowed iterator denoted wrapping return type rather passive one client code synchronous calls hasnext getnext become asynchronous message sends messages handled list actor even though part interface means concurrent usages iterators still free interesting ponder difference creating iterator inside list bestowing creating iterator outside list bestowing individual list node traverses former case getnext performed without interleaved activities actor latter case possible internal operations interleaved operations list smaller object returned concurrency sometimes desirable multiple operations object carried noninterleaved fashion purpose use atomic block construct operates actor bestowed object figure case operations actor message sends inside atomic block batched sent single message receiver case operations object guarded lock replace individual single wrapping block possible synchronise across multiple locked objects single block desired type change implicit adaptation castegren wrigstad class iterator var current node def getnext val elem elem possible interleaving messages next return elem class iterator var current node def getnext atomic val elem elem next return elem figure left right concurrency control atomic block allows client express new operations composing smaller ones situation sketched client wants access two adjacent nodes list actor without interleaving operations clients easily resolved wrapping two calls get getnext iterator used inside atomic block batch messages ensure processed back back atomic list getiterator val val list val val val return formalism explain bestow atomic use simple lambda calculus actors passive objects abstract away details unimportant describing behavior bestowed objects example leave classes actor interfaces simply allow arbitrary operations values disallowing sharing passive objects show language free syntax calculus shown figure expression variable function application message send messages sent anonymous functions executed receiving actor abstract updates passive objects actual effect formalism reasoned new object actor created new passive object bestowed current actor bestow need special atomic construct formalism modeled composing operations single message sketched end previous section statically values anonymous functions unit value dynamically identifier actor memory location passive object passive object bestowed actor type active type passive type function type unit type active type either actor type bestowed type note simplicity new bestow unit figure syntax simple lambda calculus actors bestow atomic actors without borders amnesty imprisoned state variables every passive object type every actor type every bestowed object type expressions new new tat mutate unit unit bestow unit figure static semantics maps variables types contains active types static semantics typing rules formal language found figure typing context maps variables types normal lambda calculus rules straightforward new keyword create new passive objects actors passive objects may mutated tat may bestowed activity message sends modeled sending anonymous functions run receiver receiver must active type actor bestowed object argument anonymous function must passive type thought receiver finally free variables body message must active type make sure passive objects leaked owning actors captured contains active mappings dynamically body may contain passive objects typing values straightforward dynamic semantics figure shows operational semantics language running program heap maps actor identifiers actors actor local heap actor set containing passive objects created actor message queue list lambdas run current expression evaluated actor whose current expression value may pop message message queue apply actor may step current expression possibly also causing effect heap relation denotes actor evaluating heap expression one step castegren wrigstad evaluation evaluation expressions fresh new tat mutate bestow fresh fresh new bestow figure dynamic semantics sending lambda actor prepends lambda receiver message queue results unit value sending lambda bestowed value instead prepends new lambda queue actor bestowed simply applies underlying passive object function application replaces occurrences parameter body argument mutation practice tat bestowing passive value actor creates bestowed value creating new object actor adds fresh location set actors passive objects results value creating new actor adds new actor fresh identifier heap local heap contains fresh queue empty current expression unit value handle evaluation order using evaluation context heap actors respect local heaps two different actors disjoint use denote local heap actor actor local heap message actors without borders amnesty imprisoned state dom dom dom figure rules gets local heap actor queue current expression must typable empty environment passive objects subexpressions must local heap similarly actor identifiers must actors system bestowed objects must belong local heap actor bestowed message queue messages message anonymous function taking passive argument body restrictions values current expression actor meta theory prove soundness language proving progress preservation standard fashion progress heap either evaluated one step actors empty message queues fully reduced expressions dom preservation evaluation preserves heaps properties proven hold straightforward induction main property interested language freedom actual effects passive objects show proving actor execute actor execute mutate object freedom two actors never mutate active object castegren wrigstad property simple prove using two observations makes heap actor ever access passive objects local heap local heaps actors disjoint key showing preservation first property premise rule states free variables values must active objects prevents sending passive objects actors without bestowing first sending message bestowed object always relay actor owns underlying passive object premise preservation second property simple show since local heaps grow monotonically ever extended fresh locations made observations trivial see actor heap execute must local heap another actor execute must local heap actor local heaps disjoint must different since heaps preserved evaluation programs free related work important property many systems single actor reasoned sequentially messages exchanged concurrently executed sequentially receiving actor property hold actors often rely actor isolation state one actor accessed another case concurrent updates shared state could lead breaking sequential reasoning existing techniques achieving actor isolation often based restricting aliasing example copying data passed actors relying linear types transfer ownership data bestowed objects offer alternative technique relaxes actor isolation allows sharing data without sacrificing sequential reasoning combining bestowed objects linear types straightforwand allows ownership transfer bestowed sharing actors system miller programming model based function passing rather passing data concurrent actors functions sent collections stationary immutable data called silos bestowed objects related sense sharing actually move data actors function passing model could used provide interface internal part silo implicitly relay functions passed owning silo formalism also works passing functions around abstract away unimportant details proposed programming model references bestowed objects close spirit remote references distributed programming eventual references latter case unit encapsulation actor aggregate object protected lock acts similar vat identifiable boundary identity associated interface bestowing exposing unit encapsulation safely delegate parts interface inner objects turn need internally aware kind concurrency control offered bestower actors without borders amnesty imprisoned state discussion although formal description examples focus actors bestow also works threads locks object protected lock share one internal objects requiring interaction object also goes via lock believe also straightforward extension software transactional memory future would like study combinations bestowed objects lets actor expose internal details implementation breaking encapsulation always done care leaking abstractions leads increased coupling modules lead clients observing internal data inconsistent state latter problem bestowed objects however interactions bestowed objects synchronised owning actor message queue long data always consistent messages never access data inconsistent state data inconsistent messages problem without bestowed objects sharing bestowed objects may increase contention owner message queue messages bestowed object sent owner similarly since bestowed object protected lock owner sharing bestowed objects may lead lock polled often always using locks risk introducing deadlocks believe bestowed objects exacerbate problem deadlocks caused passing bestowed object back owner easily avoided using reentrant locks accessing would require taking lock twice using locks atomic blocks similar java actors atomic block groups messages single message fairness may make sense allow atomic blocks send limited number messages possible synchronise several locked objects simply grabbing several locks synchronising several actors involved requires actors wait communicate progress actor starts finishes others canonical example atomically withdrawing depositing amount accounts two different actors interestingly accounts bestowed objects actor bank actor atomic transaction implemented message batching approach suggested paper leave future work implementation currently working implementing bestowed objects atomic blocks context encore uses active objects concurrency encore object passive active interface defined class methods defined therein may invoked thus follow formal model message passing implemented sending anonymous functions however use approach implementation bestowed objects atomic blocks extend active class implicit method perform takes function applies receiver returns result wrapped future bestowed object logically implemented object two fields owner object message send foo bestowed object translated message send perform atomic block implemented sketched end messages batched sent single message castegren wrigstad atomic foo bar perform implementation works discussed somewhat limiting allow caller react intermediate values therefore exploring alternative approach temporarily switch message queue active object one caller submit messages messages passed active object end original message queue processed first atomic block finishes active object would implicitly extended two methods override switches current message queue new one resume discards temporary queue resumes execution original queue logically translation could look like val new messagequeue atomic override val foo val foo val val baz baz resume message processed receiver stops reading regular message queue instead starts using queue provided caller rather sending messages normally caller interacts queue waiting responses necessary message processed receiver goes back reading messages normally abstracting synchronisation methods finally note connection safe type qualifier introduced kappa type system ranges actors locks immutables value safe type accessed concurrently without risk achieved depends type value runtime let type safe equivalent foo actor returning future value get blocking read future protected lock access equivalent lock unlock immutable special synchronisation needed consequently safe qualifier used express operations objects concurrency control abstracted without losing safety atomic block used atomically compose operations safe object choice concurrency control mechanism relegated runtime similarly bestowed objects internally knowledge concurrency control thus bestowed object used safe object neither object client needs knows interaction made safe conclusion actor isolation important maintain sequential reasoning actors behavior bestowing activity internal objects actor share representation without losing sequential reasoning without bloating interface atomic blocks client create new behavior composing smaller operations bestowed objects need know access safe trust safety living world actors borders actors without borders amnesty imprisoned state references agha actors model concurrent computation distributed systems series artificial intelligence mit press armstrong history erlang hopl iii brandauer parallel objects multicores glimpse parallel language encore formal methods multicore programming castegren wrigstad reference capabilities concurrency control ecoop clebsch drossopoulou blessing mcneil deny capabilities safe fast actors agere haller odersky capabilities uniqueness borrowing ecoop heather miller philipp haller normen jocelyn boullier function passing model typed distributed functional programming onward miller robust composition towards unified approach access control concurrency control thesis johns hopkins university usa modular specification verification programs berlin heidelberg srinivasan mycroft kilim actors java ecoop
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apr adaptive minimax sparse estimation interactions chenglong yuhong yang school statistics university minnesota ford hall church minneapolis usa abstract linear regression interaction effects broadly applied research fields bioinformatics social science paper first investigate minimax rate convergence regression estimation sparse linear models interactions derive matching upper lower bounds three types heredity conditions strong heredity weak heredity heredity results stronger heredity condition may may drastically improve minimax rate convergence fact situations minimax rates convergence three heredity conditions minimax rate convergence determined maximum total price estimating main effects estimating interaction effects goes beyond purely comparing order number main effects interaction effects iii three heredity conditions estimation interaction terms may dominant part determining rate convergence two different reasons exist interaction terms main effect terms large ambient dimension makes challenging estimate even small number interaction terms second construct adaptive estimator achieves minimax rate convergence regardless true heredity condition sparsity indices msc subject classifications primary secondary keywords phrases minimax rate convergence sparsity highdimensional regression quadratic model interaction selection heredity condition hierarchical structure adaptive estimation introduction data increasingly prevalent various areas bioinformatics astronomy climate science social science number variables larger sample size linear regression setting statistical estimation regression function often requires crucial conditions one common condition sparsity data generating model small portion variables important affect response variable condition sparse estimation linear regression functions variable selection well studied fruitful theoretical understandings recent decade minimax estimation regression function main effects well investigated minimax estimation interactions sparsity constraints van geer candes tao bunea zhang huang van geer van geer bhlmann bickel zhang knight raskutti rigollet tsybakov wang model selection consistency results also obtained various model selection procedures fan zhao zhang huang zou yuan fan however models main effects often adequate fully capture nature data interaction terms may necessary improve prediction performance also enhance understanding relationships among variables especially areas genetics medicine behavioristics interaction effects covariates enormous interest hierarchical constraints often imposed describe underlying structure models interaction effects marginality principle nelder effect heredity principle hamada models peixoto follow popular naming convention heredity conditions adopted chipman strong heredity weak heredity strong heredity assumes interaction term model corresponding main effects also included weak heredity requires least one main effects included practice possible compared interaction terms main effects small including modeling may beneficial perspective estimation variability thus work take consideration additional case heredity condition imposed also purpose theoretical comparison two heredity conditions many approaches proposed interaction selection categorized two types joint selection selection joint selection approach selects main interaction terms simultaneously searching possible models interactions typical way joint selection use regularization methods specially designed penalty terms example yuan introduced family shrinkage estimators incorporate hierarchical structures linear equality constraints coefficients possess selection consistency estimation consistency fixed choi regression model interactions applied adaptive penalty estimators oracle property fan hao proposed computationally efficient regularization algorithm marginality principle ramp simultaneously selects main effects interaction effects quadratic effects data also verified interaction selection consistency property lasso sensible conditions selection procedure first performs main effect selection excluding interaction terms reduce dimension variables carries joint selection reduced dimension variables computationally feasible effective example viewing sliced inverse regression likelihood perspective jiang liu suggested variable selection algorithm siri minimax estimation interactions via inverse regression able detect higher order interactions without imposing hierarchical structures hao zhang proposed two stagewise interaction selection procedures ifort iform enjoy sure screening property first stage fan proposed method named interaction pursuit incorporates screening variable selection dimensions method possesses sure screening property oracle property two stages respectively works interaction selection see zhao bien hall xue aforementioned good properties types interaction selection approaches disadvantages well joint selection usually computational infeasible insufficient storage large selection pointed hao zhang may difficult theoretically justified general conditions although many novel developments selection interaction terms described little work done estimation regression function interactions exist paper present theoretical results minimax rate convergence estimating regression function interaction terms three different hierarchical structures regardless heredity condition results show minimax rate determined maximum total estimation price main effects interaction effects heredity conditions enter minimax rate convergence terms estimation price interaction effects namely log number nonzero interaction effects number eligible candidate interaction terms different heredity conditions consequently stronger heredity condition leads possibly faster minimax rate convergence example underlying model main effects interaction terms allowed enter model strong heredity compared weak heredity seen certain situations minimax rate improved imposing strong heredity although strong heredity allows fewer eligible interaction terms two heredity conditions also perspective estimation may difference rate convergence weak heredity heredity many situations results provide complete characterization comparison minimax rates convergence three heredity conditions real applications since one know true heredity condition behind data practically best heredity condition describe data given sample size desirable construct estimator performs optimally matter three heredity conditions holds estimator adapts true heredity condition well unknown number main interaction effects obtained paper remainder paper organized follows section introduce model setup loss function heredity conditions problem section stating required assumption present main results minimax rate convergence strong heredity theoretical results minimax estimation interactions weak heredity heredity presented section section provides detailed rates convergence different heredity conditions relation sparsity indices ambient dimension sample size followed section present interesting implications detailed results section extend results quadratic models quadratic interaction effects considered section construct adaptive estimator achieves minimax rate convergence without knowledge type heredity condition sparsity indices proofs results technical tools presented appendix preliminaries model setup suppose dataset composed matrix observations covariates response vector start considering linear regression model main effects interaction effects overall coefficient vector full design matrix random noise vector known specifically coefficients main effects interaction effects respectively define matrix contains interaction terms denotes product two vectors paper focus fixed design covariates considered given goal estimate mean regression function linear combination covariates interaction terms loss function denote mean regression function denote estimated function fixed design setting focus prediction loss averaged squared error euclidean norm set index sets main effects interaction effects imain iint respectively let imain iint cartesian product index set model main effects interaction effects paper consider data generating model least two main effects one interaction effect purely convenience affect conclusions let submatrix corresponds model index corresponding least squares estimator used estimate projection matrix onto column space loss function using model denoted minimax estimation interactions heredity conditions denote space hierarchical notation subscripts vectors refer subvector consisting first elements subvector containing rest elements introduce following two vector spaces space captures strong heredity condition interaction term model corresponding main effects also included space characterizes weak heredity condition interaction model least one main effects included pointed hao zhang sign main effect coefficients invariant linear transformation covariates individually due existence interaction terms heredity conditions consequently meaningless without specification model parametrization paper stick parameterization include heredity condition considering vector space define vector number elements kak vector space define corresponding respectively note represents collection coefficients main effects interaction effects certain hierarchical constraint denotes collection linear combinations covariates coefficients throughout paper assume otherwise minimax risk may converge rate may optimal minimax risk helpful consider uniform performance modeling procedure plentiful choices modeling procedures analysis statistical problem minimax framework seeks estimator minimizes worst performance statistical risk assuming truth belongs function class minimax risk consider min max minimax estimation interactions estimators min max may refer inf sup formally speaking work assume true mean regression function hierarchical structure imposing paper use notation represent hold denote indicate order holds without use notation minimax rate convergence strong heredity assumption start stating assumption required result minimax rate convergence strong heredity paper use indicate number main effects infinity increases also allow increase sample size well sparse reisz condition src exist constants depending min min src assumption requires eigenvalues zti sparse submatrix bounded away first proposed zhang huang similar sparse eigenvalue conditions zhang raskutti condition rigollet tsybakov also related stringent restricted isometry property requires constants close candes tao assumptions standard regularization analysis like lasso dantzig selector see bickel meinshausen van geer koltchinskii references minimax rate present main result minimax rate convergence strong heredity simple estimator effective minimax upper enoughi model minimizes bound let arg residual sum squares models exactly main effects interaction effects strong heredity denoted istrong projection onto column space minimax estimation interactions design matrix lower bounding minimax risk informationtheoretical tool using fano inequality metric entropy understanding yang barron plays important role proof theorem sparse reisz condition strong heredity condition minimax risk upper bounded min max sup log log pure constant minimax risk lower bounded log log min max positive constant depends constants src assumption theorem src strong heredity condition imax rate convergence scales log log remark term log log reflects aspects estimation main effects price searching among possible models order log price estimating main effect coefficients search thus log total price estimating main effects similarly log total price estimating interaction effects remark result upper bound general require sparsity condition although may needed fast rate convergence minimax rate convergence weak heredity heredity similar results obtained weak heredity heredity minimax rate convergence still determined maximum total price estimating main effects interaction effects heredity condition changes total price estimating interaction effects may differ possibly substantially theorem sparse reisz condition weak heredity condition minimax risk order min max log log minimax estimation interactions theorem sparse reisz condition heredity condition minimax risk order log log min max comparisons insights section summarize consequences main results three scenarios integrated understanding brevity introduce following notation define quantity log total price estimating main effects interaction effects denoted respectively depends heredity condition also use notation indicate depends heredity condition let min max denote minimax risk heredity condition detailed rates convergence since minimax rate convergence depends maximum discuss cases one two quantities greater scenario main effects interaction effects sense minimax rate convergence affected heredity conditions log log always regardless heredity conditions log log depends order decide estimation price larger log log let otherwise summary given minimax risk order log log otherwise remark scenario also includes special case must minimax rate convergence standard parametric order regardless heredity conditions minimax estimation interactions scenario log exist interaction terms weak heredity quantity always less order strong heredity discuss case case log log always log log depends order decide estimation price larger terms order log log let otherwise summary given log minimax risk order log log otherwise remark term deals case inactive sense exceeds specific heredity condition example upper bound provide new information number interaction effects strong heredity thus automatically reduced subset scenario log number main effects least exponentially many main effects sense log always less terms order fact scenario results minimax rates weak heredity exactly scenario completeness still present results specifically minimax risk order interesting implications comparing results weak heredity heredity may may distinct rates convergence exists small constant log log large enough minimax estimation interactions difference weak heredity heredity perspective rate convergence estimation still remains open question different problem model identification without relationship guarantee rates convergence weak heredity heredity example log addition minimax rates weak heredity log log log contrast instead minimax rates different log log log heredity conditions affect rates convergence situations example exist main effects interaction effects scenario minimax rates convergence three heredity conditions detailed rates convergence three heredity conditions estimation interaction terms may become dominating part two different reasons price estimating interaction terms becomes higher main effect terms one number interaction terms main effect terms reason although main effect terms outnumber interaction terms ambient dimension large even estimating small number interaction terms challenging estimating main effects much rate convergence improved imposing strong heredity quantify improvement taking ratio two minimax rates convergence given ambient dimension scenario log log maximal improvement happens log minimax rate convergence strong heredity log times faster weak heredity similarly maximal improvement happens log scenario log improvement log maximal improvement happens scenario maximal improvement minimax rate weak heredity strong heredity depends ambient dimension words larger ambient dimension improvement minimax rate convergence weak heredity strong heredity similarly log equality holds active three heredity conditions maximal improvement minimax rate heredity strong heredity turns consistent log maximal improvement minimax estimation interactions pens log extension quadratic models aforementioned results consider quadratic effects quadratic interaction effects included model called quadratic model easy see rates convergence theorems still apply strong heredity weak heredity however case heredity number quadratic terms enters minimax rate assume one model extra quadratic terms need following assumption sparse reisz condition exist constants depending min min min new design matrix representing matrix contains quadratic terms next state minimax results quadratic models strong heredity weak heredity exactly condition since quadratic term one corresponding main effect term strong weak heredity require quadratic term coefficient must also coefficient similarly minimax rate convergence heredity quadratic model stays order log log heredity order becomes log log adaptation heredity conditions sparsity indices previous sections determined minimax rates convergence estimating linear regression function interactions different sizes sparsity indices heredity conditions results assume minimax estimation interactions known however practice usually prior information underlying heredity condition sparsity constraints thus necessary appealing build estimator adaptively achieves minimax rate convergence without knowledge construct adaptive estimator achieve goal consider one specific model three types models together candidate models weak ipn istrong ipn denotes full model main effects interaction effects included risk estimator worse order rank full design matrix weak slight abuse notation use istrong represent model main effects interaction effects strong heredity weak heredity heredity respectively note models appear cause problem goal estimating regression function details range model class shown choose model candidate set apply abc criterion yang model criterion value abc projection onto column space design matrix rank descriptive complexity model constant model descriptive complexity satisfies exp model descriptive complexity crucial building adaptive model let four constants set cipn log full model cistrong log log log log log ciweak log cino log minimax estimation interactions complexity assignment recognizes three types models different heredity conditions let arg abc denote model minimizes abc criterion candidate model set denote least squares estimate using model following oracle inequality theorem log worst risk abc estimator upper bounded log log sup rank full design matrix constant depends constant theorem without prior knowledge sparsity indices constructed abc estimator adaptively achieves minimax upper bound regardless heredity conditions result also indicates major difference estimation model identification estimation result able achieve adaptation respect heredity condition without additional assumption model identification although aware work addresses task adaptation unknown heredity nature seems certain much stronger assumptions consistency individual heredity condition necessary achieve adaptive selection consistency achieving adaptive model selection consistency different types conditions remains important open problem model selection theory methodology remark require assumptions relationship among variables upper bound theorem particular variables may arbitrary correlated remark order achievable use projection estimator full model thus minimax rate convergence slower order known rank design matrix plays important role determining minimax rate convergence fixed design yang rigollet tsybakov wang result together make total estimation price true model small enough upper bound improved log log remark abc estimator may practical large case stochastic search instead subset selection used implementation minimax estimation interactions remark term automatically applies lower bound whichever heredity condition since src assumption intrinsically requires larger terms order otherwise lower bound proof exceed upper bound leads contradiction give specific example appendix illustrate requirement appendix proof upper bound theorem proof recall set estimator model use ztn denote mean regression function vector row full design matrix first prove equivalently abc estimator candidate set consider src assumption assures follows model corresponding submatrix full rank thus arg min arg min arg min abc collection models main effects interaction effects models share model descriptive complexity log log cistrong abc criterion model descriptive complexity introduced near therefore abc estimator candidate set next prove upper bound since abc estimator candidate set theorem yang khi inf positive constant depends constant exists specific model projection estimator model equal consider rhs minimax estimation interactions evaluated model still denote convenience thus term bounded follows log log log log log log therefore log log thus min max max log log universal constants appendix proof lower bound theorem stating proof introduce local metric entropy two important sets aid understanding metric entropy regression function space together lemmas relation two sets metric entropy metric entropy plays central role minimax theory concepts packing covering provides way understand cardinality set infinitely many elements deriving lower bound information theoretic techniques play key role local metric entropy fano inequality shannon mutual information divergence begin introducing definition local metric entropy minimax estimation interactions definition local metric entropy given metric space let around denoted log mxa defined entropy denoted log mlocal defined maximum supremum maximum exist log mxa log mlocal max log mxa important subsets set hamming distance two vectors consider set let denote subset first coordinates fixed let denote another subset interaction effect exists following two lemmas metric entropy subsets needed proof exists subset cardinality lemma pairwise hamming distance less exp log points subset greater proof proof presented appendix lemma exists subset cardinality pairwise hamming distance less exp log points subset greater proof proof similar lemma proof since log suffices prove risk function class monotonicity minimax reduces proof case similarly suffices prove minimax estimation interactions recall coefficient space interest mean regression function space convenience let denote regression functions coefficents respectively let radius around ball radius around underlying regression function without loss generality assume square root empirical loss used measure distance two functions prove following two cases separately case log log consider subset product two vectors lemma exists subset hsub exp log pairwise hamming distance elements within hsub greater set hsub hsub hsubp exist hsub since also mapping hsub hsub thus hsub pairwise elements hsub greater let hsub src assumption minimax estimation interactions subset let follows fsub hsub pairwise distance terms functions implies entropy less lower bounded log log less log mlocal log yang barron minimax log risk lower bounded log log log constant depends case log log consider subset following arguments conclude minimax lower bounded log log log constant depends log notice log similarly log log together fact lower bounds two cases minimax risk lower bounded log log thus desired lower bound holds remark one way interpret imposition src assumption indeed constant divergence two joint densities joint distribution response variable fixed design parameterized respectively see let row joint density exp parameter distance minimax estimation interactions proof lemma since main effects fixed fix first let denote collection points within hamming distances follows cardinality bounded upper bound since main effects fixed point need pick positions interaction effects different remaining interaction effect positions gives possible choices positions coordinates positions take values thus desired upper bound follows let subset consider collection points within hamming distance element strictly less inequality implies set induction create set hamming distance two elements exceeds next introduce one useful inequality minimax estimation interactions thus log log desired result follows appendix proof theorem proof proofs similar arguments strong heredity slight differences prove upper bound heredity instead consider weak minimizes residual sum model arg squares models main effects interaction effects weak heredity model descriptive complexity thus log different strong heredity case ciweak log abc criteria models defined arguments proof used prove lower bound weak heredity consider set hweak two important subsets instead hweak hweak similar metric entropy results two subsets derived fashion lemmas arguments proof appendix proof theorem proof thep upper bound heredity consider model arg model descriptive complexity cino log log abc criteria models defined lower bound heredity consider set hno minimax estimation interactions two important subsets instead hno hno similar metric entropy results two subsets derived fashion lemmas arguments proofs appendix proof theorem model descriptive complexity term plays fundamental role model selection theory barron cover barron yang wang since considering models interaction terms model descriptive complexity reflects comprehension model complexity total number parameters detailed designation descriptive complexity usually depends class models interest instead interpreting code length description length describing model index one also treat exp prior probability assigned model bayesian viewpoint proof candidate set represented union candidate sets three heredity conditions fstrong fweak fno fstrong ipn istrong fweak ipn iweak fno ipn ino exists specific model fstrong projection estimator model equal also projection onto full design matrix still denote two models minimax estimation interactions ipn respectively follows khi inf inf khi log hipn log rank full design matrix first inequality follows second inequality follows fstrong third inequality results evaluation ipn two terms bounded follows log log log log log log log log log log log log log therefore max log log constants depend constant thus desired minimax upper bounded follows fweak fno replacing fstrong quantity instead greater minimax estimation interactions log log log log log weak heredity log log log log log heredity different constants affect conclusion terms order following arguments proof strong heredity desired results follow underlying heredity condition weak heredity heredity appendix example src satisfied simplicity let consider example regression mean function includes one main effect term corresponding src assumption exist constants depend rpn design matrix matrix contains main effects assume first columns linearly independent denote zrz submatrix rank suppose submatrix satisfies src assumption assume kzi purpose illustration set let collection columns satisfy know cos angle two vectors thus cos cos cos cos otherwise less violates src assumption since minimax estimation interactions implies means two elements pairwise distance greater less well known entropy unit ball order log denote ball radius let constant exists positive log log log since set ball radius cardinality satisfies log log covering number packing number closely related inequality thus log log log log implies elements src assumption src assumption thus long satisfied src assumption requires must hold pair columns case lower bound log theorems apply references barron massart risk bounds model selection via penalization probability theory related fields barron cover minimum complexity density estimation ieee transactions information theory bickel ritov tsybakov simultaneous analysis lasso dantzig selector annals statistics bien taylor tibshirani lasso hierarchical interactions annals statistics bunea tsybakov wegkamp aggregation gaussian regression annals statistics candes tao dantzig selector statistical estimation much larger annals statistics chipman bayesian variable selection related predictors canadian journal statistics revue canadienne statistique choi zhu variable selection strong heredity constraint oracle property journal american statistical association fan variable selection via nonconcave penalized likelihood oracle properties journal american statistical association fan kong interaction pursuit feature screening selection arxiv preprint hall xue selecting interacting features highdimensional data computational statistics data analysis hamada analysis designed experiments complex aliasing journal quality technology united states minimax estimation interactions hao feng zhang model selection high dimensional quadratic regression via regularization journal american statistical association hao zhang interaction screening ultrahighdimensional data journal american statistical association hao zhang note high dimensional linear regression interactions american statistician jiang liu variable selection general index models via sliced inverse regression annals statistics knight asymptotics estimators annals statistics koltchinskii dantzig selector sparsity oracle inequalities bernoulli sliced inverse regression dimension reduction journal american statistical association zhong zhu feature screening via distance correlation learning journal american statistical association fan unified approach model selection sparse recovery using regularized least squares annals statistics meinshausen recovery sparse representations data annals statistics nelder reformulation linear models journal royal statistical society series general peixoto hierarchical variable selection polynomial regression models american statistician raskutti wainwright minimax rates estimation linear regression ieee transactions information theory rigollet tsybakov exponential screening optimal rates sparse estimation annals statistics van geer deterministic lasso van geer generalized linear models lasso ann statist van geer bhlmann conditions used prove oracle results lasso electron statist wang paterlini gao yang adaptive minimax regression estimation sparse journal machine learning research yang model selection nonparametric regression statistica sinica yang barron determination minimax rates convergence annals statistics yuan joseph zou structured variable selection minimax estimation interactions estimation annals applied statistics zhang nearly unbiased variable selection minimax concave penalty ann statist zhang huang sparsity bias lasso selection linear regression annals statistics zhang analysis convex relaxation sparse regularization journal machine learning research mar zhao rocha composite absolute penalties family grouped hierarchical variable selection annals statistics zhao model selection consistency lasso journal machine learning research nov zou yuan composite quantile regression oracle model selection theory ann statist
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estimation directed information yonathan murin department electrical engineering stanford university usa nov moriny abstract report studies estimation directed information measure twoem random process based estimation framework detailed derivations two estimators provided two estimators differ metric based found facilitate estimation measure assumed observed sequences jointly markovian order generally known method also based principle estimating observed sequences presented exhaustive numerical study shows discussed estimators perform well even relatively small number samples thousands moreover shown discussed estimators capable accurately detecting linear well causal interactions introduction detection estimation causality relationships two random processes fundamental problem many natural social sciences task particular challenging many scenarios one good underlying statistical model considered process fields neuroscience financial markets meteorology etc scenarios desirable use estimator causal influence two observed timeseries common approach quantifying causal influence two dates back seminal work granger said influence given past past helps predicting future samples approach indeed general common formulation granger causality assumes obey linear structure therefore follow approach mentioned possible alternative information theoretic measure directed information closely related transfer entropy functional report discuss estimation two sequences estimated used measure causal influence random processes underlying observed note deterministic function joint density underlying random processes therefore estimated first estimating local joint densities using densities estimate approach taken proposed use kernel density estimator kde estimating local densities suggested estimate local densities using note work presented scenarios fail detect quantify causal influence pair conclusions also hold measure correlation integrals hand estimation method discuss current report based principle delving technical details estimating functional emphasize estimating statistical functionals common assume underlying process stationary ergodic smooth rest report build upon assumptions without verifying validity rest report organized follows section discuss estimation differential entropy estimation approach extended estimating mutual information functional section estimating discussed section numerical study presented section notation denote random variables rvs upper case letters realizations corresponding lower case letters use notation denote sequence denote random processes using boldface letters denote sets calligraphic letters denotes set real numbers denotes probability density function pdf continuous log denotes natural basis logarithm finally use denote differential entropy mutual information defined estimation differential entropy introduce concept estimation information theoretic measures first discussing estimation differential entropy detailed discussion regarding methods including estimating differential entropy divergence provided popular approach estimating information theoretic functionals method first density locally around data points estimated functional estimated via empirical averaging sec specifically let independent identically distributed samples pdf consider estimating differential entropy let local estimation density around ith sample recalling differential entropy defined log see estimator given log several popular techniques estimating local density kernel density estimation correlation integrals report focus estimation algorithms based principle presenting estimation strategy provide several definitions let defined defined let sort sorted version vector ascending order define sort sorted vector distances samples particular denotes distance unless otherwise stated report assume fixed relatively small number independent range assuming uniform local density small environment around samples assumption implies density smooth used estimate let denote euler gamma function let denote volume unit dimensions given since ball radius samples total approximate local density via substituting obtain following entropy estimator log work kozachenko leonenko showed simple estimator biased suggested following estimator log log log log log digamma function remark case dependent chosen function converges log required obtain consistent estimator hand fixed correction term log crucial consistency next discuss estimation mutual information entropy mutual information consider two rvs rdx rdy observing pairs joint density interested estimating fixing estimating entropy terms via one obtains following consistent estimator consistency follows consistency estimator denoted note estimator entropy term estimated separately different distance thus even though estimator consistent since estimators coupled finite number samples bias motivated work kraskov grassberger ksg presented modification estimator empirically showed modification improves performance higher accuracy number samples finite following refer estimator main idea behind modify estimators individual entropy terms correlation estimator joint entropy term higher leading smaller bias interpretation recently presented let denote distance pair define indicator function let defined similarly note number samples within distance distance measured thus places limitation distance follows using definitions setting ksg estimator given iksg log note cdx cdx cdy max hence ksg estimator estimates individual entropy via log log cdx log sample dependent hand estimation joint entropy term identical one work showed consistent derived order bias remark inherent bias estimator uses radius around sample point since max follows one individual terms exactly distance term case see fig illustration observation thus letting assuming lies around bias estimator order inherent bias partially addressed second estimator introduced eqn estimator uses instead yet using requires approximations thus none two estimators presented uniformly better refer reader fig detailed discussion regarding approximations used apart second estimator motivated idea using sample dependent work proposed different method tackle inherent bias discussed remark idea use ball instead ball used choice distance distance see fig however euclidean norm used namely new relationship derived number points relationship formulated thm essentially states following approximation cdx approximation motivates estimating log via log log log cdx log plugging estimation results gov entropy estimator log log cdx log log finally obtain estimator recall cdx cdx using estimator estimate joint entropy estimators individual entropy terms one obtains log log comparing log cdx cdx cdx cdy cdx log log viewed correction term using euclidean norm finally note consistency stated thm directed information definitions background let arbitrary random processes let sequences directed information defined follows defining differential entropy definition implies independent given viewed quantifying causal influence sequence sequence therefore surprising contrast symmetric directed information rate processes defined lim provided limit exists make following assumptions regarding processes random processes assumed stationary ergodic markovian order observed sequences stationarity assumption implies statistics considered random processes constant throughout observed sequences note formally speaking random processes stationary order ensure existence rate practical perspective required sequences stationary ensure causal influence change observed sequences ergodicity assumed ensure observed sequences truly represent underlying processes finally markovity assumption common modeling systems finite memory example used markov models analysis neural recordings used markov models financial modeling social networks dynamics formulate assumption markovity order observed sequences via use simplifying assumption dependence past samples order definitions estimation methods defined sequel easily extended two different orders moreover implicitly assume given independent reflects setting simultaneously measured entropy first sample exists following conditional entropy exists assumptions required mathematically insure rate exists equal simple expression depends finite memory length moreover assumptions prevent degenerate case deterministic deterministic relationship one scenarios discussed assumptions lemmas imply exists equal view given following interpretation given past sequence namely much past sequence namely helps predicting next sample observed function markov order one interested estimator unknown must estimated observed sequences estimating markov order reasonable assumption observed finite memory thus obey markov model order estimated data possible approach estimating choose value facilitates best prediction future samples see references therein specifically let finite set candidate markov orders estimated value set minimizes loss function predicting next sample previous samples proposed use prediction method use prediction next sample based past samples approach ignores dependency account dependency propose predict next sample based past samples let prediction function predicts measure quality prediction use distance thus model order estimated via argmin figure prediction illustration prediction assuming expectation averages everything random approximated via averaging note prediction method used fact proposed use ensemble predictors order increase prediction power yet comes cost much higher computational complexity moreover numerical study indicates markov order small efficient predictor significantly complicated models regression based terms hand large fixed suffers curse dimensionality section performs poorly note stated sec number samples required accurate estimation grows exponentially markov order follows increasing viewed increasing state space therefore many practical settings number samples limited focus settings relatively fig illustrates procedure estimating estimator first finds tuples nearest tuple tuples response variable fig responses used predict resulting loss note instead averaging one use weighted averaging based distances procedure repeated every tuple note search among tuples overlap current tuple calculated loss values averaged resulting procedure repeated value minimize average loss declared next discuss estimation measure assuming markov order correctly estimated estimating directed information simplify notation let denote past denote past using notation write hold one either observed sequences decimate mating see discussion ksg estimator begin estimator extends approach used estimate estimator presented estimating measure let denote distance tuple following refer distance recalling dimensions note calculated space dimension similarly individual entropies estimated using distance calculated largest space resulting estimators individual entropy terms given log log log log log log log log log log log log combining entropy estimators obtain similarly ksg estimator joint entropy estimated using estimator see entropy terms estimated using sample dependent expressions see gov estimator second estimator extends approach used estimate following steps leading obtain following estimators log log log log log log log log combining entropy estimators obtain log statistical significance estimated since estimated finite number samples one may want assess statistical significance estimation lack statistical significance may imply causal influence underlying random processes estimated values due either noise estimation error case one may choose set estimated zero estimating known methods quantifying statistical significance via respective sec known estimation continuous alphabet sequences alternative method evaluating statistical significance via bootstrapping procedure spirit specifically idea randomly shuffle causal interactions destroyed changed repeat shuffle procedure times estimate denotes shuffled sequence since construction destroys causal influence new estimated values assumed taken causal influence statistical significance determined resulting parameter finally note main drawback bootstrapping procedure significant increase computational complexity since applying procedure amounts multiplying computational complexity factor least common value jvhw estimator ksg gov estimators allow continuous input alphabet different approach first quantize observed sequences bins order estimate empirical probability mass functions pmfs estimate pmfs estimator developed jiao venkat han weissman jvhw based optimal estimation entropy discrete distributions specifically estimator uses relationship independently estimate entropy terms main advantages proven optimality estimating entropy terms discrete distributions well linear time complexity contrast universal estimators detailed hand number available samples small quantizing input sequences lead significant performance degradation see numerical study section quantize observed sequences one may use optimal scalar quantizer numerical study showed cardinality discrete alphabet large enough quantization yields performance similar naive quantization based assumed probability observing samples large magnitude negligible evaluate statistical significance estimated values use bootstrapping procedure discussed previous section note samples work derived asymptotic result may used number samples small numerical study section examine performance discussed estimators several simulated scenarios also consider estimation via toolbox motivated fact widely used quantify causal influence particular generated multivariate process equal twice consider linear well scenarios scenarios taken focus regime order thousands contrast focused case relatively small values motivated practical applications number samples limited simulation results indicate discussed estimators ksg gov achieve roughly accuracy estimator presented order magnitude less samples linear interaction let gaussian rvs independent generate sequence via observe causally influence causal influence shown appendix given log cosh measured bits fig depicts mean estimated dis versus value values estimated samples mean calculated independent different random generator seeds trials plots fig average standard deviation estimator average standard deviation jvhw estimator average standard deviation ksg estimator average standard deviation gov estimator statistical significance estimated values evaluated using approach discussed section estimates used stated sec estimators yield significant estimations trials estimator yield significant estimations trials ksg gov estimators yield significant estimations trials ksg gov estimators yield significant estimations trials respectively significance jvhw estimator gradually increases range true values relatively large estimators yield significant estimations trials theory jvhw jvhw ksg ksg gov gov average average theory jvhw jvhw ksg ksg gov gov figure average estimates versus interaction four quantization levels used jvhw estimator ksg gov estimators comparing estimation results fig estimation results reported fig one observe curves similar ksg gov versus kde estimator yet fundamental difference estimators fig number samples fig number samples thus considered scenario achieve accuracy ksg gov estimators require order magnitude less samples compared kde estimator explore performance discussed estimators larger repeated experiment specified results depicted fig observing fig note accuracy ksg gov estimator significantly higher compared fig moreover standard deviations estimated values smaller order magnitude hand one observe apparent bias bias due inaccurate estimation markov order noise level significantly higher signal level estimation method discussed section estimates true markov order verify observation fig present average estimated ksg estimator several alternatives markov order estimation approach discussed section corresponding curve denoted baseline approach proposed estimator used estimate markov order based past samples ignoring past samples corresponding curve denoted ragwitz fixing markov order correct order according setting corresponding curve denoted setting markov order corresponding curve denoted observed method performs much worse method discussed section fact curve corresponding ragwitz method almost identical curve generated setting setting true order leads almost errors comparing theoretical values finally setting works better small values problem theory baseline baseline ragwitz ragwitz average average theory jvhw jvhw ksg ksg gov gov figure average estimates versus interaction six quantization levels used jvhw estimator ksg gov estimators average estimated jvhw ksg gov estimators average estimated ksg estimator different markov order estimation methods estimating markov order challenging yet significant performance degradation observed larger values quadratic interaction next consider quadratic dependency let gaussian rvs independent generate sequence via similar model studied sec similarly section consider two scenarios expect increase monotonically expect observe shape fig similarly fig causal influence note calculating theoretical values model seems intractable fig depicts mean estimated dis versus value samples used estimated values mean values reported fig calculated independent trials similarly linear scenario plots average standard deviation estimator average standard deviation jvhw estimator average standard deviation ksg estimator average standard deviation gov estimator except estimator statistical significance also similar linear scenario observed expected estimated monotonically increases moreover results similar fig exception significantly smaller number samples used estimation versus finally fig indicates jvhw estimator able capture causal influence although estimated values smaller estimated jvhw jvhw ksg ksg gov gov average average jvhw jvhw ksg ksg gov gov figure average estimates versus interaction four quantization levels used jvhw estimator ksg gov estimators estimators hand expected indicated able capture causal influence follows based linear model obeys quadratic one next discuss two coupling maps discussed noisy map first study model consists unidirectional coupling via map let let generate sequence via next let gaussian rvs independent generate sequence via gain parameter sets estimated last samples sequences first note parameter controls strength coupling two sequences moreover indicates causally influences influence increase hand based expect causal influence discussed noiseless scenario namely fully synchronized two sequences indistinguishable thus case causal flow information zero fig considers almost noiseless indicates ksg estimator indeed estimates increasing causal influence moreover causal influence drops almost zero increased beyond indicating almost set exactly zero order violate assumption prevent numerical instabilities estimating jvhw jvhw ksg ksg gov gov average average jvhw jvhw ksg ksg gov gov figure average estimates versus interaction four quantization levels used jvhw estimator ksg gov estimators deterministic relationship result full correspondence figs fig also indicates ksg estimator correctly infers negligible causal influence values observed fig curve corresponding gov estimator biased jvhw estimator misses drop causal influence fig consider case observed ksg estimator results similar case exception directions significantly larger zero exactly matches desired behavior indicated figs setting gov estimator seems less biased jvhw still misses decrease causal influence finally note fails infer significant causal influence sigmoid coupling discuss scenario drives sigmoid function sec defined sigmoid exp let generate sequence via cos sigmoid gaussian rvs independent indicates causally influences influence increase hand based expect causal influence jvhw jvhw ksg ksg gov gov averagedi figure average estimates versus interaction four quantization levels used jvhw estimator ksg gov estimators fig depicts average estimated versus observed four estimators increases note curve ksg gov jvhw average corresponding estimator somewhat surprising interaction clearly linear examining average observed gov jvhw find causal influence exist according sec hand ksg estimator estimates negligible amount causal influence thus ksg estimator one follow expected results indicated figs references palus vejmelka bhattacharya causality detection based approaches time series analysis physics reports vol granger investigating causal relations econometric models methods econometrica vol massey causality feedback directed information proc int symp inf theory waikiki usa kramer directed information channels feedback dissertation swiss federal institute technology wibral vicente lindner transfer entropy neuroscience directed information measures neuroscience understanding complex systems janzing balduzzi scholkopf quantifying causal influences annals statistics vol malladi kalamangalam tandon aazhang identifying seizure onset zone causal connectivity inferred using directed information ieee jour sel topics sig vol sabesan good tsakalis spanias treiman iasemidis information flow application epileptogenic focus localization intracranial eeg ieee trans neural syst rehabil vol cover thomas elements information theory edition interscience wang kulkarni universal estimation information measures analog sources foundations trends communications information theory vol moon rajagopalan lall estimation mutual information using kernel density estimators physical review vol grassberger procaccia measuring strangeness strange attractors physica vol olver lozier boisvert clark nist handbook mathematical functions cambridge university press wang volumes 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propagation social networks proc int conf machine learning atlanta usa murin goldsmith aazhang estimating memory order electrocorticography recordings submitted ieee transactions biomedical engineering ragwtiz kantz markov models data simple nonlinear time series predictors delay embedding spaces physical review vol hastie tibshirani friedman elements statistical learning springer jiao venkat han weissman minimax estimation functionals discrete distributions ieee transactions information theory vol may murin kim parvizi goldsmith sozrank new approach localizing epileptic seizure onset zone submitted plos comput murin kim goldsmith tracking epileptic seizure activity via information theoretic graphs asilomar conference signals systems computers pacific grove usa barnett sheth mvgc multivariate granger causality toolbox new approach inference journal neuroscience methods vol diks degoede general nonparametric bootstrap test granger causality institute physics publishing global analysis dynamical systems jvhw shannon entropy renyi entropy mutual information estimator http accessed jiao permuter zhao kim weissman universal estimation directed information ieee transactions information theory vol kontoyiannis skoularidou estimating directed information testing causality ieee transactions information theory vol soltani inferring signaling structures brain via directed information doctoral thesis stanford university diamandis murin goldsmith ranking causal influence financial markets via directed information graphs annual conference information sciences systems princeton usa submitted ishiguro otsu lungarella comparison nonlinear granger causality extensions systems physical review vol mapping strange attractor communications mathematical physics vol
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message passing combinatorial optimization siamak ravanbakhsh thesis submitted partial fulfillment requirements degree doctor philosophy department computing science university alberta siamak ravanbakhsh abstract graphical models use intuitive methods graph theory implicitly represent dependencies variables large systems model global behaviour complex system specifying local thesis studies inference discrete graphical models algebraic perspective ways inference used express approximate combinatorial problems investigate complexity reducibility various inference problems part organizing inference hierarchy investigate tractable approximations subset problems using distributive law form message passing quality resulting message passing procedure called belief propagation depends influence loops graphical model contribute three classes approximations improve loopy graphs loop correction techniques survey propagation another message passing technique surpasses settings iii hybrid methods interpolate deterministic message passing markov chain monte carlo inference review existing message passing solutions provide novel graphical models inference techniques combinatorial problems three broad classes constraint satisfaction problems csps satisfiability coloring packing set dominating independent set optimization counterparts clustering problems hierarchical clustering modularity optimization iii problems permutations including bottleneck assignment graph morphisms alignment finding symmetries bottleneck traveling salesman problem many cases show message passing able find solutions either near optimal favourably compare today approaches memory grandparents iii acknowledgements many helped survive grow graduate studies first would like thank supervisor russell greiner learned teaching value common sense academia also granting rare freedom research years chance learn many great minds alberta elsewhere pleasure working david wishart brendan frey jack tuszynski also grateful committee members dale schuurmans csaba mohammad salavatipour external examiner cristopher moore valuable feedback enjoyed many friendly conversations colleagues babak alipanahi nasimeh asgarian trent bjorndahl kit chen andrew delong jason grant bret hoehn sheehan khan philip liu alireza makhzani rupasri mandal james neufeld christopher srinivasa mike wilson others thankful friends amir amin arash arezoo azad babak fariba farzaneh hootan hossein kent kiana maria mariana meys meysam mohsen mohammad neda niousha pirooz sadaf saeed saman shaham sharron stuart yasin yavar good time canada finally like thank family specially reihaneh tare kindness constant support acknowledge help mental stimulation playful minds internet reddit stackexchange funding support computing science help desk alberta innovates technology futures alberta innovates center machine learning compute canada contents abstract acknowledgements contents list tables list figures general notation xii introduction contributions inference message passing representation complexity reducibility problem inference inference hierarchy single marginalization complexity general inference classes complexity hierarchy inference reductions marginalization integration marginalization reduces integration integration reduces marginalization reductions reduces reduces approximate inference belief propagation computational complexity limits message passing tractable factors sparse factors factors factors large number constraint factors inference optimization friends message passing relaxation families loop corrections short loops long loops message dependencies short loops message experiments grids regular graphs survey propagation semirings semirings decomposition integral new semiring new integral marginals counting survey propagation messages particles markov chain monte carlo algorithm gibbs sampling hybrid methods perturbed belief propagation combining operators experiments perturbed survey propagation combinatorial problems constraint satisfaction phase transitions random csps revisiting survey propagation flavours decimation computational complexity perturbed survey propagation csp satisfiability coloring problem dominating set set cover pitfalls decimation complexity vii clique problem independent set sphere packing binary variable categorical variable efficient sphere packing hamming distance optimization variations csps clustering facility location problem hierarchical clustering spanning steiner trees problem problem modularity maximization potts model clique model representation simplified message update augmentation experiments permutations matching permanent complexity arbitrary graphs traveling salesman problem augmentative approach simplified messages augmentation experiments viii using pairwise factors graph matching problems isomorphism subgraph monomorphism supermorphism graph homomorphism reduction csps finding symmetries graph alignment conclusion future work references list tables correspondence comparison different methods comparison different modularity optimization methods summary solutions combinatorial problems list figures variables circles factors squares belief propagation illustration distributions used loop correction example generalized loop correction method accuracy different levels difficult various loop correction methods versus accuracy different problem sizes various loop correction methods survey propagation comparison marginalization accuracy gibbs sampling perturbed set solutions simple problem comparison perturbed benchmark csps schematic view phase transitions rcsps condensation phase perturbed behaviour comparison different methods example induced set cover example message passing hamming distances distance factors white squares black experimental results clustering experimental results clustering facility location example maximum modularity clustering message passing matching example application augmentative message passing solve tsp experimental results tsp using message passing various benchmark instances experimental results bottleneck tsp quality marginals integral graph endomorphism relaxed conditions conjecture example finding approximate symmetry using endomorphism marginals example sparse graph matching message passing xii introduction many complicated systems modeled graphical structure interacting local functions many fields almost independently discovered graphical models used bioinformatics protein folding medical imaging spectroscopy pedagogy trees regulatory networks neuroscience formation associative memory neuroplasticity communication theory low density parity check codes statistical physics physics dense matter theory image processing inpainting reconstruction denoising compressed sensing robotics particle filters sensor networks social networks natural language processing speech recognition artificial intelligence artificial neural networks bayesian networks combinatorial optimization thesis concerned application graphical models solving combinatorial optimization problems broadly put seeks optimal assignment discrete set variables brute force approach infeasible see decomposition offered graphical model model complex system consider joint distribution binary variables naive way represent would require table entries however variables conditionally independent dependence structure forms tree exactly represent joint distribution using values operations marginalization require computation time linear original size reduced local computation form message passing structure tree case reduces cost linear new exponentially smaller size turns even dependency structure loops use message passing perform approximate inference moreover approach problem inference algebraic point view contrast variational perspective local computation two perspectives extent residuals different origins research statistical physics statistical study physical systems boltzmann distribution relates probability state physical system energy often decomposed due local interactions studies often interested modeling systems thermodynamic limit infinite variables average behaviour study random ensembles inference techniques origin cavity methods often asymptotically exact assumptions importantly studies reduced inference optimization notion free energy variational approach contrast graphical models community emerged study knowledge representation reasoning uncertainty advances characterized attention theory computation logic interest computational opposed analytical solutions motivated study approximability computational complexity invention inference techniques belief propagation efficient exact tree structures also studies lead algebraic abstractions modeling systems allow local computation common foundation underlying two approaches information theory derivation probabilistic principles logical axioms leads notions entropy divergences closely linked physical entropy free energies physical systems less abstract level shown inference techniques communication attempting minimize approximations free energy another exchange ideas two fields study critical phenomenon random constraint satisfaction problems computer scientists physicists satisfiability heart theory computation important topic investigate reasoning hand study critical phenomena phase transitions central statistical physics disordered systems culminated variational analysis lead discovery survey propagation constraint satisfaction significantly advanced solving random satisfiability problems despite convergence variational algebraic perspectives extent complementary variational approach extend beyond log probabilities algebraic approach justify application message passing graphs loops although briefly review variational perspective thesis mostly concerned algebraic perspective particular rather study phase transitions behaviour set solutions combinatorial problems concerned finding solutions individual instances part starts expressing general form inference proposes novel inference hierarchy studies complexity chapter also show problems reducible others introduce algebraic structures make efficient inference possible general form notation reductions proposed chapter used later chapters chapter studies forms approximate inference first introducing belief propagation considers problems intractably large number factors factors large cardinality solutions problems study different modes inference optimization review alternatives convergent procedures convex linear programming relaxations inference classes inference hierarchy standard message passing using belief propagation guaranteed exact graphical structure loops optimization perspective variational perspective also led design approximate inference techniques account short loops graph different family loop correction techniques account long loops taking message dependencies account chapter reviews methods introduces novel loop correction scheme account short long loops resulting accurate inference difficult instances message passing loopy graphs seen fixed point iteration procedure existence loops means may one fixed point therefore alternative loop correction way incorporate fixed points performed also message passing procedure known survey propagation next section chapter introduces survey propagation novel algebraic perspective enables performing inference set fixed points another major approach inference offered markov chain monte carlo mcmc techniques minimal review mcmc final section chapter introduces hybrid inference procedure called perturbed belief propagation interpolates belief propagation gibbs sampling show technique outperform belief propagation gibbs sampling particular settings part thesis uses inference techniques derived first part solve wide range combinatorial problems review existing message passing solutions provide novel formulations three broad classes problems constraint satisfaction problems csps clustering problems combinatorial problems permutations particular chapter use perturbed belief propagation perturbed survey propagation obtain performance random satisfiability coloring problems also introduce novel message passing solutions review existing methods sphere packing cliquecover several optimization counterparts applying perturbed belief propagation graphical representation packing problem able compute long optimal nonlinear binary codes large number digits chapter proposes message passing solutions several clustering problems modularity optimization shows message passing able find solutions moderate instances problems also review previous approaches hierarchical clustering also related graphical models minimum spanning tree steiner tree chapter deals combinatorial problems permutations first reviewing existing graphical models matching approximation permanent graph alignment introducing two novel message passing solutions versions traveling salesman problem bottleneck tsp study graph matching problems including graph isomorphism monomorphism homomorphism graph alignment approximate symmetries particular study graph homomorphism show graphical model generalizes several problems including hamiltonian cycle clique problem coloring show graph homomorphism used surrogate isomorphism find symmetries contributions acknowledgment results thesis joint work supervisor greiner researchers detail algebraic approach inference presented loop correction ideas published perturbation schemes csp presented performing inference first suggested brendan frey christopher srinivasa many related ideas including reductions presented joint paper finally augmentation scheme tsp modularity maximization discussed contribution thesis including published work follows generalization inference problems graphical models including inference hierarchy limit distributive law tree structures theorems propositions claims complexity inference including inference general commutative semirings unified treatment different modes inference identification key properties significance inverse operator several settings including loop correction schemes survey propagation equations reduction inference inference simplified form loop correction markov networks generalization incorporate short loops regions novel algebraic perspective survey propagation perturbed perturbed application constraint satisfaction problems augmentation inference intractably large number constraints formulation several combinatorial problems including vertex cover packing model packing hamming distances problem clique model modularity optimization tsp bottleneck tsp general framework study graph matching including subgraph although previous work claim address problem note formulation monomorphism rather isomorphism study message passing homomorphism finding approximate symmetries graph alignment diverse set penalties part inference message passing part thesis first studies representation formalism hierarchy inference problems reducibilities underlying algebraic structure allows efficient inference form message passing graphical models chapter viewing inference different lights procedures allow better approximations chapter chapter representation complexity reducibility chapter use simple algebraic structure commutative semigroup express general form inference graphical models end first introduce representation formalize inference section section focuses four operations defined summation multiplication minimization maximization construct hierarchy inference problems within pspace problems class hierarchy belong complexity class encounter new inference problems establish completeness problems lower levels hierarchy complexity classes section augment simple structures two properties obtain message passing commutative semirings also observe replacing semigroup abelian group gives normalized marginalization form inference inquiry show inference commutative semiring postpone investigation message passing next chapter section shows inference problems introduced far reducible others problem inference use commutative semigroups define graphical model represents also define inference graphical model idea using structures semigroups monoids semirings expressing inference long history approach based factorgraphs commutative semigroups generalizes variety previous frameworks including markov networks bayesian networks forney graphs hybrid models influence diagrams valuation networks particular combination semigroups consider generalizes plausibility feasibility utility framework pralet explicitly reduced problem inference graphical models mentioned many main difference approach keeping framework free semantics decision chance variables utilities constraints often associated variables factors operations without changing expressive power notions later associated individual inference problems help interpretation definition commutative semigroup pair set binary operation associative commutative commutative monoid commutative semigroup plus identity element every element inverse often written commutative monoid abelian group associativity commutativity properties commutative semigroup make operations invariant order elements general properties vital one may define inference starting example examples semigroups set strings concatenation operation forms semigroup empty string identity element however semigroup commutative set natural numbers summation defines commutative semigroup integers modulo addition defines abelian group set intersection operation defines commutative semigroup identity element set natural numbers greatest common divisor defines commutative monoid identity fact semilattice commutative semigroup given two commutative semigroups two sets cartesian product also commutative semigroup let tuple discrete variables domain let denote subset variable indices tuple variables indexed subset factor function subset variables range factor definition pair collection factors collective range poly magma generalizes semigroup require associativity property identity element inference graphical models also extended use magma definition elements ordered parenthesized avoid ambiguity order pairwise operations set avoid unnecessary complications confine treatment commutative semigroups chapter representation complexity reducibility polynomial representation possible evaluate polynomial time commutative semigroup closure compactly represents expanded joint form note connection set factors commutative semigroup range factors conditions definition necessary sufficient compactly represent evaluate expanded form polynomial time stronger condition ensure factor compact representation poly means explicitly expressed array conveniently represented bipartite graph includes two sets nodes variable nodes factor nodes variable node note often identify variable index connected factor node set also index use denote neighbours variable factor node factor graph set also use denote markov blanket node example figure shows variables factors assuming min expanded form represents min assume variables binary hypercube one assignment corner also assume factors count number nonzero variables complete assignment easy check expanded form min marginalization operation shrinks expanded form using another commutative semigroup binary operation inference combination expansion one marginalization operations computationally intractable due exponential size expanded form definition given function commutative semigroup closure marginal def problem inference figure variables circles factors squares short means compute one perform operation set assignments tuple think tensor marginalization performing operation axes set result another tensor function call marginal marginalization dimensions denote marginal instead call integral define inference problem sequence marginalizations expanded form definition inference problem seeks closure collective range factors commutative semigroups partition set variable indices polynomial representation poly chapter representation complexity reducibility note refer potentially different operations belongs different semigroup call inference problem integration denoting inquiry otherwise call marginalization constant sized always enough ensure polynomial representation size individual may grow exponentially see claim following call expansion semigroup marginalization semigroup example going back example shaded region figure shows partitioning variables use define following inference problem max min min associate problem following semantics may think factor agent payoff agent depends subset variables adversarial variables environmental chance variables controlled variables query variables inference problem query seeks maximize expected minimum payoff agents without observing adversarial chance variables assuming adversary makes decision observing control chance variables example probabilistic graphical model defined using expansion semigroup often marginalization semigroup expanded form represents unq normalized joint probability whose marginal probabilities simply called marginals replacing summation marginalization semigroup max seeks maximum probq ability state resulting integration problem maxx known maximum posteriori map inference alternatively adding second marginalization operation summation get marginal map inference max max object interest negative energy product expansion semigroup replaced instead sum marginalization semigroup use exp semigroup log integral case function change marginalization semigroup min integral minimum energy corresponding map example probabilistic graphical model ising model ferromagnetism model extensively studied physics mainly model phase transition magnets inference hierarchy model consists binary variables denoting magnet spins arranged nodes graph usually grid cayley tree energy function hamiltonian associated configuration joint form jij variable interactions denoted called local field defines factor jij local fields define local factors depending type interactions call resulting ising model ferromagnetic jij setting neighbouring variables likely take similar values jij kind interactions allowed particular ferromagnetic interactions comparable frequency model called class problem shows interesting behaviours completely understood see studied phenomena materials important connections difficult inference problems including combinatorial optimization problems two well studied models spin glass edwardanderson models model defined grid interactions grid model complete graph inference hierarchy often complexity class concerned decision version inference problem definition decision version inference problem asks question integral given produce hierarchy inference problems analogy polynomial counting arithmetic hierarchies define hierarchy assume following definition two consecutive marginalization operations distinct marginalization index sets moreover log call marginalization operation polynomial marginalization poly defining required factor polynomially computable building hierarchy require operations semigroup polynomially computable well end consider set rational numbers note automatically eliminates semigroups involve operations exponentiation logarithm closed operations consider summation product minimization maximization chapter representation complexity reducibility always inference problem enforce first two conditions therefore impose restriction following use language identify inference problems arbitrary set factors example refers inference problem sense rightmost token language product identifies expansion semigroup rest tokens identify marginalization semigroups given order therefore minimal language exactly identifies inference problem information affects computational complexity inference problem specified language whether marginalization operations polynomial exponential define five inference families families associated outermost marginalization operation definition family inference problems sum similarly associated product minimization maximization family inference problems last marginalization polynomial log regardless define inference classes family problems class computational complexity hierarchy exhaustive includes inference problems four operations sum min max product whenever integral polynomial representation see claim moreover inference classes disjoint family parameterized subscript two sets inference class family number marginalization operations set indices exponential set indices polynomial marginalizations example identifies decision problem min partition assume since three marginalization operations first second marginalizations exponential third one polynomial since constant therefore since exponential summation inference hierarchy problem belongs class alternatively use different values linearly grow corresponding inference problem becomes member remark note arbitrary assignments necessarily define valid inference class example require index larger moreover values compatible inference class example inference class member notational convenience inference class notation invalid inference hierarchy equate empty set means inference class rather definition ignore inference problems product appears marginalization semigroups following claim explains choice claim prod inference query exponential representation proof claim states product appears marginalization operations marginal integral become large longer represent polynomial space show integration problem idea show exponential representation marginal query see integral exponential representation consider simplified form result inference last marginalization step product grows exponentially recall hierarchy defined operations since constant size say size representation using binary scheme exponential define base members families def def def def sum min max prod def sum sum min min max max initial members family identify expansion semigroup sum identifies exception contains three inference let denote union corresponding classes within families treat specially case marginalization operation polynomial log violates conditions definition inference problem chapter representation complexity reducibility define inference family members recursively adding marginalization operation problems inference class marginalization polynomial new class belongs family set updated accordingly alternatively outermost marginalization exponential depending new marginal operation min max sum new class defined member case last marginalization summation set updated adding exponential marginalization poly def sum def min def max def adding polynomial marginalization def poly min max sum single marginalization inference classes hierarchy one marginalization min min max max sum sum max min max max min max min min sum prod sum min sum max review problems prove complete conp respectively starting proposition inference proof show inference problems provide algorithms sum sum short asks sum assignments sum factors easy see factor value counted times summation therefore inference hierarchy rewrite integral new form involves polynomial number terms therefore easy calculate min min similar max max short min min query seeks minimum achievable value factor easily obtain seeking range factors reporting minimum value polynomial time widely studied known decision version npcomplete reduction satisfiability show inference also proposition decision version inference asks maxx mini npcomplete proof given easy verify decision problem decision belongs show reduce inference problem satisfiable iff value unsatisfiable otherwise simply define one factor per clause satisfies clause number less one otherwise construction value maxx one iff original sat problem satisfiable otherwise less one reduces means problems fact complete complexity class contrast problems conp class decision problems instances result polynomial time verifiable witness proof note changing decision problem complexity classes problems family reversed problems become problems become conpcomplete among members known easy show result inference proposition decision problem mini class problems polynomially solvable using turing machine acceptance condition majority computation paths accept chapter representation complexity reducibility proof see mini enumerate assignment calculate mini polynomial time path accepts iff mini accept iff least paths accept given matrix problem calculating permanent perm set permutations corresponding decision problem show completeness enough reduce problem computing matrix permanent inference graphical model problem computing permanent reduced inference graphical models however isomorphic min therefore problem computing permanent matrices reduces inference complexity general inference classes let denote complexity class inference class hierarchy obtaining complexity class problems use following fact also used polynomial hierarchy pnp pconp fact pnp pconp oracle means adding polynomial marginalization problems get complexity class following gives recursive definition complexity class problems inference note definition complexity class similar recursive definition members class equations theorem complexity inference classes hierarchy given recursion conpf npf ppf base members defined equation belong proof recall definition factor graph ensures evaluated polynomial time therefore base members complexity base members see proposition prove completeness complexity classes beyond first level hierarchy assert membership inference hierarchy use classes base induction assuming complexity classes correct show correct consider statements one one complexity members adding inference problem known complexity requires turing machine enumerate xjm possibilities call oracle reduced xjm clamped reject iff calls oracle rejects means conpa equation also making another assumption expressed following claim claim inference classes complexity fact evaluated polynomial time means contains one inference class exactly one following cases correct constructing hierarchy assume two consecutive marginalizations distinct current marginalization minimization contains single class inductive hypothesis ensures problems complexity class completes proof claim complexity members adding max inference problem known complexity requires turing machine enumerate xjm possibilities call oracle reduced accept iff calls oracle accepts means npa argument similar claim ensures equation contains single inference class complexity members adding inference problem known complexity requires turing machine enumerate xjm possibilities call oracle reduced accept iff majority calls oracle accepts means ppa fact evaluated polynomial time means chapter representation complexity reducibility despite fact different since closed complement means ppa ppa recursive definition complexity equation remains correct complexity members adding marginalization inference problem known complexity requires turing machine deterministically enumerate xjm possibilities polynomial time time call oracle reduced accept calculation means three possibilities case since ppnp ppconp recursive definition complexity equation remains correct example consider inference equation decision version problem member also includes max max complexity class according equation nppp however marginalmap also known complete nppp suppose polynomial constant belongs complexity ppp turing machine enumerate polynomial time call oracle see accept calls oracle accepts rejects otherwise reduced factor variables fixed example also hints rationale behind recursive definition complexity class inference class hierarchy consider inference family toda theorem interesting implication hierarchy theorem states hard polynomial hierarchy means min max min max inference inference hierarchy arbitrary constant number min max operations appears inference inference hierarchy complexity hierarchy restricting domain min max become isomorphic logical respectively true false considering restriction inference hierarchy two operations express quantified satisfiability qsat inference graphical model let factor disjunction min max max min min adding summation operation express stochastic satisfiability generalizing constraints disjunctions represent quantified constraint problem qcp qsat stochastic sat qcps pspace class problems solved turing machine polynomial space therefore show inference inference hierarchy pspace follows inference hierarchy pspacecomplete well theorem inference hierarchy proof theorem prove problem show pspace problem reduces already saw qsat reduces inference hierarchy difficult show inference hierarchy contained pspace let inference problem hierarchy simply iterate values nested loops using recursion let index marginalization involves moreover let ordering variable indices algorithm uses notation demonstrate procedure using nested loops note loop individual domains xik rather xjm track temporary tuples qik space complexity remains polynomial chapter representation complexity reducibility input output loop query domain zin xin loop xin loop end zin xin xin end xin xin end algorithm inference pspace inference definition inference based expansion operation one marginalization operations assume single marginalization operation polynomial time inference still generally possible however assume expansion operation distributive marginalization loops exact polynomial time inference possible definition commutative semiring combination two commutative semigroups two additional properties identity elements moreover annihilator distributive property dealing reals means inference mechanism efficient inference using distributive law seen simple example instead calculating min using fact summation distributes minimization may instead obtain result using min requires fewer operations example following examples commutative semirings max max min max ordered set min min true false semiring natural numbers greatest common divisor least common multiple lcm gcd symmetric semiring many semirings isomorphic log defines isomorphism also easy show semiring isomorphic semiring inference problems example different properties indirectly inherited commutative semirings example operation min also max choice function means implication sum semiring min max replace max required recover using polynomial time argxj another example since operations inverses field availability inverse operation abelian group important implication inference expanded form equation normalized may inquire normalized marginals def def normalized joint form deal case integral evaluates annihilator special case division annihilator may also means working normalized expanded form normalized marginals always example since abelian groups sumsum product inference normalized marginals inference means minxj however inference since max abelian normalized marginals defined chapter representation complexity reducibility apply identity annihilator commutative semiring define constraints definition constraint factor whose range limited identity annihilator expansion iff forbidden iff permissible constraint satisfaction problem csp inference problem semiring factors constraints note allows definition csp commutative semiring idea using different semirings define csps studied past however implication inference commutative semirings ignored theorem inference commutative semiring randomized reduction proof prove inference semiring randomized polynomial reduction deterministically reduce unique satisfiability usat inference problems semiring usat promise problem asks whether satisfiability problem promised either zero one satisfying assignment satisfiable valiant vazirani prove polynomial time randomized algorithm usat implies reduction consider set binary variables one per variable given instance usat clause define constraint factor satisfies clause otherwise means satisfying assignment usat iff definition instance unsatisfiable integral instance satisfiable single instance therefore integral evaluates therefore decide satisfiability usat performing inference semiring relying properties identities satisfying assignment recovered using decimation procedure assuming access oracle inference semiring example inference semiring true false xor factor disjunction form called asks whether number sat solutions even odd corollary theorem randomized reduction indeed case recall monoid semigroup identity existence identity property semiring reductions find useful use notation identity function condition def cond cond true cond false min min max intended semiring function clear context reductions several inference problems commutative semirings reducible section reviews reduction marginalization integration general commutative semirings use reduction obtain approximate message dependencies performing loop corrections section equation introduce procedure reduce integration finding normalized marginals procedure called decimation reduces sampling marginalization problem sampling distribution known almost difficult integration see chapter constraint satisfaction reduced sampling therefore marginalization section introduce perturbed message passing scheme perform approximate sampling use solve csps recent work perform approximate sampling finding map solution perturbed particular type noise added factors approximate integration also recently reduced map inference section see inference obtained limiting cases inference respectively section reduces inference also sequence csps therefore inference reduction gives powerful procedure solve minmax problems use part solve bottleneck combinatorial problems contrast type reduction various modes inference many studied reductions different types examples special forms binary variables pairwise interactions constant degree nodes planar form example sanghavi show integration reducible maximum problem however since pairwise binary represent maximum problem see section means integration reduced problem pairwise binary model reductions part motivated fact restrictions restricted allows efficient inference example possible calculate integral planar ising model see example polynomial time absence chapter representation complexity reducibility local fields complexity loop correction method study section grows exponentially degree node therefore may beneficial consider reduced factorgraph iii factors pairwise factors satisfy certain metric property polynomial algorithms obtain exact integral using marginalization integration section shows arbitrary commutative semirings reduction marginalization integration vice versa marginalization reduces integration fixed assignment subset variables evidence reduce factors intersection accordingly def identity function defined equation new factor graph produced clamping factors manner effectively accounted evidence marginalization integration performed reduced use similar notation integral marginal new factor graph recall problem integration calculating obtain marginals integration reduced reductions claim proof reductions normalize values get defined equation integration reduces marginalization assume access oracle produce normalized marginals equation show calculate making calls oracle note marginals normalized integral trivially given start given normalized marginal variable fix arbitrary value reduce factors according equation repeat process marginalization clamping times variables fixed point denotes subset variables fixed step including refers new marginal note require step fix different variable call assignment invalid annihilator semiring want avoid division annihilator using equations easy show valid assignment always exists therefore unable find valid assignment means let denote final joint assignment produced using procedure proposition integral original given inverse defined according proof first derive equation conditional normalized marginals semirings defines inverse claim semiring normalized joint form arrive equality first note since multiply chapter representation complexity reducibility sides see claim get divided sides moved term left right second step apply repeatedly get chain rule semiring xin equivalent simply substituting definition equation get procedure incremental clamping known decimation variations typically used two objectives recovering map assignment max marginals assuming maxproduct semiring instead arbitrary one picks argxj max producing unbiased sample distribution assuming semiring sample reductions objective appears various fields particularly building robust models uncertain adversarial settings context probabilistic graphical models several objectives different inference semiring previously studied also see section combinatorial optimization may refer relation maximization minimization dual combinatorial objectives corresponding linear programs may refer settings due uncertainty problem specification reductions part see several problems studied class bottleneck problems formulated using semiring instances problems include bottleneck traveling salesman problem problem bottleneck assignment problem edmonds fulkerson introduce bottleneck framework duality theorem relates objective one problem instance objective dual problem intuitive example duality cut separating nodes cut minimum maximum weight path path minimum maximum weight hochbaum shmoys leverages triangle inequality metric spaces find constant factor approximations several problems unified framework common theme majority heuristics bottleneck problems relation objective csp establish similar relation within context reducing inference problem original inference csp see section reduced equation particular since use inference solve resulting csp call reduction reduction inference reduces show inference reduces although contrast reduction next subsection polynomial time reduction first make simple observation inference let denotes union range factors value belongs set fact assignment show manipulate factors original produce new factors domain inference former corresponds inference later lemma two sets factors identical domains identical solutions argx min max argx min max chapter representation complexity reducibility proof assume different argx min maxi argx min maxi let denote corresponding values claim max max max max simply follows condition lemma case one assignments optimal assignment alternative assignment lower maximum factors lemma simply states matters solution relative ordering let ordering elements let denote rank ordering define reduction theorem argx min argx min max reduction proof first note since monotonically increasing function rank elements range rank range using lemma means argx min max since argx min max definition max argi max simplicity assuming instance single assignment case multiple assignments correspondence proof instead starts assumption assignment first different assignments second reductions follows max max therefore argx min max argx min equality combined equation prove statement theorem alternative approach use inverse temperature parameter objective objective low temperature limit lim argx min argx min max reduces recall denote union range factors reduce original problem csp using following reduction definition problem argx min max given def normalizing distribution defines csp iff satisfying assignment moreover gives number satisfying assignments following theorem basis reduction theorem let denote solution corresponding value maxi satisfiable particular unsatisfiable always probability define def chapter representation complexity reducibility proof satisfiable enough show since indicator functions rhs evaluate showing satisfiable towards contradiction assume satisfiable let denote satisfying assignment using definition implies however means maxi means value theorem enables find assignment solving sequence csps let ordering starting satisfiable hand satisfiable using binary search need solve log csps find solution moreover search upper lower bounds optimal solution latest unsatisfiable latest satisfiable reduction however finding assignment otherwise showing assignment exists general instead use incomplete solver may find solution csp satisfiable failure find solution guarantee unsatisfiability using incomplete solver lose lower bound optimal however following theorem states increase value number satisfying assignments increases making potentially easier solve proposition partition function number solutions proof recall definition maintain lower bound one able correctly assert unsatisfiability reductions means solution related ability solve gap increases potentially becomes easier solve chapter approximate inference belief propagation naive approach inference commutative semirings normalized version equation construct complete array using tensor product perform however number elements exponential number variables loop free use distributive law make inference tractable assuming marginal interest form tree root starting leaves using distributive law move inside define messages leaves towards root follows equation defines message variable factor closer root similarly equation defines message factor variable closer root distributive law allows moving domain outside inside equation way moves place give analogous message starting leaves calculating messages towards root obtain marginal belief propagation figure figure shows direction messages sent variable factor nodes order calculate marginal grey region root node product incoming messages fact assume subset variables factors within variables root set incoming messages produces marginal example consider joint form represented figure problem calculating marginal shaded region move inside obtain chapter approximate inference term factors summation corresponding example message computational challenge however also decompose message using distributive law simplify based incoming messages variable nodes procedure known belief propagation sometimes prefixed corresponding semiring even though guaranteed produce correct answers tree cases performs surprisingly well applied fixed point iteration graphs loops case loopy graphs message updates repeatedly applied hope convergence contrast trees messages leaves root calculated message update applied update messages either synchronously asynchronously update schedule play important role convergence numerical stability operator inverse messages normalized use indicate normalization according mode inference pii general graphs approximations equation functionals cast message updates operator subset incoming messages use functional notation belief propagation presenting algebraic form survey propagation section another heuristic often employed damping messages often improves convergence applied loopy graphs damping parameter used partially update new message based old one alternative one may use expensive form geometric damping appears power apply damping either messages currently similar several ideas explore thesis damping heuristic proved utility applications lacks theoretical justification computational complexity time complexity single message update equation save computation variable large number neighbouring factors none message values equal annihilator zero inverse defined derive marginals produce messages reduces cost calculating messages leaving variable call type update update note since max abelian ordered set allow type variablesynchronous update motivates using reduction inference time complexity single message update equation however see section sparse factors allow much faster updates moreover cases reduce calculating messages leave particular factor time section call type synchronized update update chapter approximate inference limits message passing observing application distributive law semirings natural question ask use distributive law polynomial time inference graphical models inference problems higher levels inference hierarchy general inference problem one marginalization operation answer question motivated fact loops exists scheme may become powerful approximation technique one marginalization operations natural assumption using distributive law expansion operation distributes marginalization operations sum distributes min max consider simplest case three operators distributes integration problem partition order apply distributive law pair need able commute operations require specified consider simple case involving two binary variables applying equation simple case require following theorem leads immediately negative result theorem tractable factors implies direct application distributive law tractably exactly solve inference problem one marginalization operation unfeasible even tree structures limitation previously known marginal map inference min max operations interesting property regard similar operations min max min max max min however slightly change inference problem pure assignments xjl xjl distribution assignments mixed strategies result celebrated minimax theorem min max operations commute min max max min mixed strategies property enabled addressing problems min max marginalization operations using procedures example ibrahimi solve variation inference message passing procedures operate graphical models game theory graphical games also rely property tractable factors applicability graphical models discrete optimization problems limited size number factors section review large order factors allow efficient message passing focusing sparse factors used part solve combinatorial problems section introduce augmentation procedure similar cutting plane method deal large number constraint factors sparse factors formulation many interesting combinatorial problems involves sparse factors either factor involves large number variables variable domains large cardinality factors able significantly reduce time complexity calculating messages efficient message passing factors studied several works context inference classes confine discussion factors used part application sparse factors common vision many image labelling solutions problems image segmentation stereo reconstruction operate using priors enforce chapter approximate inference ilarity neighbouring pixels image processing task usually reduced finding map solution however pairwise potentials insufficient capturing statistics natural images therefore employed simplest form sparse factor combinatorial applications potts factor factor assumes domain variables tabular form across diagonal easy see allows marginalization equation performed rather another factor similar form inverse potts factor ensures fact factor constant plus matrix allows inference factors used bottleneck tsp section another class sparse factors class cardinality factors factor defined based number values gail proposes simple method refer factor factor use similar algorithms factors alternative linear clique potentials potetz lee authors propose assuming variables domain marginalization scheme general family factors called linear clique potentials nonlinear sparse factors larger values larger efficient methods evaluate sum pairs variables using auxiliary variables forming binary tree use fast fourier transform reduce complexity factors log see references completeness provide brief description efficient message passing factors inference factors since variables binary convenient assume messages normalized calculate factors normalize deriving assume least variables adjacent factor nonzero extensively use assumption message equation becomes tractable factors summation subsets least members calculate follow procedure except factor replaced assume therefore sufficient variables nonzero note equation sum iterates size least factors large summation contains exponential number terms fortunately use dynamic programming perform update basis recursion dynamic programming starting variable either zero one summation decomposed using similar recursion dynamic program reuses terms calculated factors convenient work normalized messages moreover computing message also normalize recall objective calculate min assume constraint factor satisfied since violated identity function evaluates see equation first case neighbouring variables factor minimum obtained assume neighbouring variables smallest rest zero remaining neighbouring variables need therefore need find smallest incoming messages rest messages zero due normalization setting letting identify set smallest incoming chapter approximate inference messages factor given index smallest incoming message excluding similar procedure give updates small constant obtain smallest incoming message time order requires log computations incur negligible additional cost calculating outgoing messages factor simultaneously update large number constraint factors consider scenario exponentially large number factors represent hard constraints see definition ask whether possible find feasible solution considering small fraction constraints idea start graphical model corresponding computationally tractable subset constraints obtaining solution constraints using augment model set constraints violated current solution process repeated hope might arrive solution violate constraints augmenting model constraints although theoretically guaranteed work experimental results suggest efficient practice general idea extensively studied term cutting plane methods different settings dantzig first investigated idea context tsp gomory provided elegant method identify violated constraints context finding integral solutions linear programs since used also solve variety nonlinear optimization problems context graphical models sontag jaakkola also use cutting plane method iteratively tighten marginal polytope enforces local consistency marginals see section order improve variational approximation interested augmentation process changes inference problem rather improving approximation inference requirements cutting plane method availability optimal solver often solver procedure identify violated constraints moreover operate real domain hence term plane however message passing much faster finding approximate map assignments structured optimization problems motivates using augmentation context message passing sections use procedure approximately solve tsp respectively despite losing guarantees make cutting plane method powerful augmentative inference optimization message passing several advantages first message passing highly parallelizable moreover directly obtaining integral solutions much easier find violated constraints note cutting plane method combinatorial problems operates fractional solutions whose rounding may eliminate guarantees however due assignments cutting plane methods require sophisticated tricks find violations example see application cutting plane tsp inference optimization variational approach concerned probabilities therefore section limited operations real domain variational approach inference expressed argbp min approximation true distribution inverse temperature see example formulated terms desired marginals expanding definition substituting equation equation becomes argbp min argbp min log log log log equation removed log partition function log log depend means minimum equation log appears quantity minimized equation known variational free energy two terms expected energy term def log entropy term log different families representations terms marginals produces different inference procedures generalized method inference retrieved limit chapter approximate inference ence argbp min argbp min log lim energy term linear therefore optima corner probability simplex reproducing map solution defining get form argbp min log observe using second parameter inference also lim argbp min argbp min log max log due linearity objective optima extreme points probability simplex retrieve update equations divergence minimization equation present reparametrization reparameterize using marginals general form holds commutative semiring abelian group proposition operator semiring inverse write def inverse exponentiation operator defined times proof proof use exactness trees substitute marginals equation assume ties solution inference optimization equation substituted messages denominator sages according equation used definition inverse cancel denominator intuitively denominator simply cancelling double counts since counted nominator denominator removes one friends rewriting equation ring energy minimization get argbp min replacing variational log log energy term equation exact quantity minimized known bethe free energy constraints equations ensure marginals consistent sum one following lead yedidia heskes showed stable fixed points local optima bethe free energy optimization approximates minimization equation two ways marginal constraint ensure local consistency general guarantee even joint probability marginals exists local consistency conditions marginal polytope polytope marginals realizable join probability bethe entropy exact loopy using method lagrange multipliers enforce local consistency constraints setting derivatives equation zero recovers updates optimization view inference inspired many chapter approximate inference inference techniques convex entropy approximations convergence guarantees message passing relaxation relaxation problem seeks marginals argbp min integral solution guaranteed optimal identical equation taking zero temperature limit lim bethe free energy equations convex entropy approximation ensures message passing solution recovers linear programming solution moreover replacing summation maximization corresponds temperature limit resulting convex message passing produces convex message passing agrees relaxations conditions ties beliefs general interest recovering solutions message passing retain optimality guarantees benefiting speed scalability message passing stems exploitation graphical structure one may also interpret convex variations replicating variables factors keeping corresponding messages identical replicates obtaining message updates number replicates allowed take rational values parisi introduced similar trick estimation partition function using replica trick another notable variation approximate map inference performs block coordinate descend space duals equation mplp guaranteed converge often able recover solution finally dual primal decomposition methods minimize factors separately combine estimates way agrees iteration families rephrasing variational inference equation argbp min max log loop corrections terms marginals enforcing marginal consistency constraints obtain following relaxation argbp min surprisingly resembles reduction inference equation relaxation factor equation claim equation lower bounds objective moreover integral optimal assignment proof integral solution corresponds following optimization problem argx min argx min max exact inference objective therefore integral obtain optimal solution hand relaxing integrality constraint optimality guarantee solution worse integral solution corresponding value lower bound complements upper bound obtain using combination reduction incomplete solver perturbed section used assess optimality solution extensively studied inference problem inference hierarchy section marginal map inference particular variational formulation inference substitutes entropy term equation conditional entropy loop corrections section first review methods account short loops section show perform loop correction taking account message dependencies section section introduce loop correction method benefit types chapter approximate inference loop corrections producing accurate marginals techniques used directly estimate integral approximation techniques take message dependencies account applied estimation marginals short loops consider general class methods improve inference loopy graphical model performing exact inference regions contain small loops earliest methods performs exact inference computation cost grows exponentially size largest region tree width regions form tree messages passed regional intersections algorithm still popular applications involve certain class graphs exact result required graphs low tree width extension junction tree junction graph method removes requirement regions form tree proxy two regions subset intersection rather whole intersection one still requires regions contain particular variable form tree similar ideas discussed name cluster graphs inspired connection bethe free energy belief propagation see section yedidia proposed generalized minimizes kikuchi approximation free energy cluster variational method entropy approximation obtained region collection connected variables set factors participating factor depends variables included region build cvm one starts predefined top outer regions factor included least one region add intersection two regions including variables factors recursively sub inner added region connected immediate parent reparameterizes terms marginals regions counting number region ensures variable factor counted number recursively defined formula inner regions ancestors region making distinction general region graph cvm accurately loop corrections similar substituting reparametrization equation variational free energy minimization equation get argbp min log known kikuchi approximation free energy constraints equation ensure marginals consistent across overlapping regions solving constraint optimization using method lagrange multipliers yields set recursive equations known generalized equations approximation exact loops without restrictions choice regions generalizes method well general construction region graph requires counting numbers regions variable factor belong sum different criteria choice regions see also long loops graphical models correlation variables methods insufficient complexity grows exponentially number variables region therefore necessarily inefficient account long loops graph class methods reducing correlations methods based conditioning subset variables clamped remove correlations formed paths include example consider markov network form cycle fixing single variable reduced factor graph becomes tree therefore allows exact inference several works investigate sophisticated ideas performing better inference clamping subset resulting theoretical guarantees closely related idea raoblackwellization collapsed mcmc see section hybrid approach inference particles represent partial assignment variables inference rest variables performed using deterministic method deterministic inference method used calculate value partition function possible joint assignment variables collapsed collapsed particles sampled accordingly process general form expensive one could reduce cost depending structure network loop calculus chertkov expands free energy around bethe approximation one term per generalized loop graph since number loops grows exponentially number variables expansion provide practical solution chapter approximate inference attempts made make method practical truncating loop series original loop series proposed binary valued pairwise factors generalized arbitrary even another class approximate inference methods perform loop correction estimating message dependencies graphical model methods particularly interesting directly compensate violated assumption corresponding independent set incoming messages benefit clarity confine loop correction equations section generalization next section markov networks see versions although previous works loop corrections concerned inference present loop corrections general commutative semiring operation inverse group particular means loop corrections may used class inference rewrite update equations markov figure left shows messages part markov network markov network tree assumption independent valid messages summarize effect separate node however graph loops use denote message dependencies access function could easily change message update equation since clear estimate follow different path instead estimate cavity distribution denote simply joint marginal markov blanket making cavity removing variable neighbouring factors however since missing information dependence messages degree freedom individual marginals inaccurate factor without affecting message passing procedure degree freedom essentially equivalent freedom initialization messages following show resulting message passing first write updates different form note rather conventional way defining formulation original equation markov network graph loop size two loop corrections figure left messages markov network ideal way dependencies taken account right marginal extended markov blanket node message dependencies markov blanket node define extended markov blanket markov blanket plus see figure right write marginals using equation simplifies assume given dependency messages form means equation enforcing marginal consistency retrieve message update similar equation incorporates dependency chapter approximate inference messages easy verify update reduces updates equation uniform dependency messages method mooij similar however interpretation apply updates factor graphs extend idea perform overlapping regions connected variables section pass messages one region outer boundary another region main computational cost loop correction methods estimating message dependencies use clamping perform task remove immediately depending factors graph approximate marginal reduction integrab tion see section note obtained way contains dependencies also individual marginals absence node however since messages updates perform corrections joint probability need divide individual marginals message dependencies short loops section presented loop correction methods improve loopy considering interactions within small clusters variables thus taking small loops within clusters account previous section showed account dependency messages thus taking correlations account section introduce generalization performs types loop correction basic idea form regions perform exact inference regions take short loops account however performing message passing regions introduce method perform loop correction messages start defining region set connected variables note definition different definition region methods specifies set variables factors let markov blanket region let def region neighbour neighbourhood iff markov blanket intersects note different messages exchanged neighbourhoods message region example figure shows set neighbouring regions indexed region receives overlapping messages four regions example message overlaps message well therefore simply writing loop corrections figure top seven regions domain messages sent region domain message region region bottom message shows overlapping messages combined prevent terms factors inside incoming messages equation variables message similar section construct track however one per region construction similar cluster variational methods start original recursively add intersections added connected immediate parent figure shows region discussions around particular drop def let top regions consisting incoming messages region interest formula equation gives counting number top regions counting number one downward pass starting top regions calculates belief average beliefs parents chapter approximate inference example figure belief average beliefs marginalized counting number require operator semiring inverse recall power operator def semirings operator corresponds exponentiation times product respectively also rational numbers define average def avg using denote parents region downward pass avg top regions incoming messages let factors defined subset example figure product pairwise factors edges figure downward pass belief analogous equation semiring product factors inside region beliefs messageregions inside double counts taken account example assuming ring inverse product division real domain point also introduce estimate message dependencies equation generalize update equation one last issue resolve define effective message different since directly use previous iteration include directly update instead use message region calculate effective loop corrections message effective message defined equation efficiently calculated upward pass message starting parents lowest regions update belief obtained downward pass equation using new beliefs children set children message upward pass new beliefs top regions gives effective messages example example figure bottom assuming ring since two layers write effective message form loop correction call generalized loop correction glc generalizes correction schemes sections following theorem makes relation generalized explicit theorem markov networks regions partition variables generalized fixed point particular construction also fixed point glc using uniform message dependencies experiments section compares different variations generalized loop correction method glc sumproduct ring well cluster variational method cvm section loop correction lcbp exactly account short loops tree expectation propagation treeep method also performs kind loop correction cvm use doubleloop algorithm slower generalized better convergence see proof methods applied without damping stop method maximum iterations change probability distribution messages less chapter approximate inference report time seconds error method average absolute error single variable marginals setting report average results random instances problem experimented grids random lcbp glc used without information message dependencies initial cavity distribution estimated via clamping cavity variables experiments full means message dependencies estimated uniform means set uniform distribution loopcorrection regions use glc denote case regions selected overlap overlapping clusters form used example full refers setting message dependencies contains overlapping loop clusters length factor appear loops forms cluster form clusters used cvm grids experimented periodic ising grids example general smaller local fields larger variable interactions result difficult problems sampled local fields independently interactions figure summarizes results grids different values also experimented periodic grids different sizes generated sampling factor entries independently figure compares computation time error different methods grids sizes range regular graphs generated two sets experiments random graphs nodes degree variables used ising model local fields couplings independently sampled figure shows time error different values figure shows time versus error graph size nodes results suggest taking long short loops account significantly improve accuracy inference cost computation time fact plots figure show trend time versus accuracy suggests taking short long loops account almost independently improved quality inference survey propagation semirings semirings survey propagation first introduced message passing solution satisfiability later generalized general csp arbitrary inference problems several evaluations based implementation libdai inference toolbox survey propagation semirings semirings ising grid ising model graph figure average accuracy spinglass ising grids ising model different values variable interactions sampled local fields sampled chapter approximate inference ising grid ising model graph figure time error ising grids ising models local field interactions sampled standard normal method graph points representing ising model different size variables works offer different interpretations generalizations survey propagation propose generalization based notions extends application arbitrary commutative semirings derivation closely follows generalizes variational approach montanari way algebraic approach using commutative semirings generalizes variational derivation fixed point iteration procedure one fixed points may converge alternatively messages initialized properly may converge one fixed points equations take fixed points account algebraic perspective accounting fixed points using third operation particular require also distribute forming second commutative semiring refer new semiring semiring let fixed point denote set fixed points fixed point corresponds approximation denote using functional form emphasize dependence approximation messages recall original problem domain assignments expanded form approximately performed case survey propagation domain assignments integral evaluates particular survey propagation semirings semirings table correspondence belief propagation survey propagation domain expanded form marginalization integration factors new expanded form assignment messages algebraic perspective efficiently performs second integral using fixed points table summarizes correspondence derivation requires abelian group every element inverse requirement invertablity need work normalized messages section introduce another variation simply counts fixed points relaxes requirement decomposition integral writing normalized equations section hid normalization constant using sign explicitly define normalization constants local integrals defining unnormalized chapter approximate inference messages based normalized version def def def def def def def def update also functional form case local integrals simply integral unnormalized messages marginals define functional product messages vice versa def def theorem loops abelian group global integral decomposes local integrals words proof proof build tree around root node connected one factor since tree node always exists send messages leaves towards root back leaves message give integral contains nodes factors node using noting root connected exactly one factor global integral hand following relation also corresponding survey propagation semirings semirings message substituting equation get summing equations substituting equation get similarly equation using integration substitution equation equation simply recursive integration tree integral node equation reduced integral unrolling recursion see simply product messages towards root equation tells global integral different therefore equation completely expand recursion global integral let restrict factor higher variable tree closer root similarly let variable closer root write global integral proposition shows local integrals written terms local integrals interest chapter approximate inference substituting equations equation get equations theorem proof proposition definition equation last step used equation similarly second statement proposition new semiring decomposition integral theorem means factored form therefore set variables three different types factors corresponding different terms decomposition represent figure shows simple corresponding new one variable per message original three types factors discussed survey propagation simply belief propagation applied new using new semiring messages exchanged variables factors simplify messages substitution keep two types messages use denote two types messages messages exchanged two types factors namely since third type factors always connected two variables simplify role message survey propagation semirings semirings figure part left corresponding right variables messages original graph three type factors iii arrows suggest message updates simplified two type messages exchanged factors type update get cases assuming messages consistent satisfy equations original note using normalized message update normalization factor hidden using sign possible assumed inverse simplify update using following proposition proposition chapter approximate inference proof definition equation last step used equation similarly second statement proposition term proposition appear equation terms local message integrals given equation enforce enforcing locally message updates updates new combining constraint simplification offered proposition gives message updates identity function semiring true false message functional possible messages variable factor however updating messages identity functions ensure messages locally satisfy equations taken account another difference updates equation messages single argument new local integrals either depend example variational approach survey propagation comes two variations entropic energetic readers familiar variational derivation express relation algebraic approach according variational view partition function entropic log partition function semiring entropic inverse temperature parameter parisi parameter easy see corresponds algebraic approach survey propagation semirings semirings limits corresponds max hand limit amounts ignoring corresponds counting see section energetic different sense log ground state energy corresponds max limits inverse temperature parameter equivalent min min taking algebraic view choose operations domains instance implication algebraic view variations applied domain complex numbers new integral marginals use theorem time approximate integral using local integral messages marginal message corresponding message see figure note message variable connected two factors factors already contained calculating one messages moreover marginals messages recover marginals marginals denote simply need enumerate combinations messages produce particular marginal counting survey propagation previously required operator inverse decompose integral local integrals moreover consistent decomposition integral semiring previously shared lift requirements discarding integrals altogether means semiring could completely distinct semiring abelian group setting particularly interesting semiring real domain expansion operation different expansion operation expanded form would evaluate integral even without loops chapter approximate inference resulting integral simply counts number fixed points marginals marginals given equation approximates frequency particular marginal original survey propagation equations successful solving satisfiability correspond counting applied semiring example interestingly problems discrete domains messages take values range factors ordered set closed min max operations counting message discrete distribution possible messages means counting survey propagation semring computationally tractable contrast counting applied real domains tractable case message distribution uncountable set practice counting interesting remains tractable case corresponds counting applied semiring case factors constraints domain messages true false algebraic perspective extends set tractable instances example show counting used count number fixed points applied semiring messages particles contrasting properties stochastic deterministic approximations make general hybrid method desirable reviewing basics mcmc section discuss approaches message passing introduce hybrid inference method combines message passing gibbs sampling section discussions section limited inference markov chain monte carlo markov chain monte carlo mcmc technique produce samples target distribution exploring markov chain constructed probable areas visited often markov chain stochastic process current state independent history given previous state homogeneous markov chain transition kernel case starting arbitrary distribution least tmix assumptions irreducibility probability reaching states starting arbitrary state aperiodicity chain trap cycles messages particles transitions chain tmix given set particles sampled estimate marginal probabilities expectation given transition kernel following condition known detailed balance identifies stationary distribution means left eigenvector eigenvalue eigenvalues less one mixing time tmix chain depends second largest eigenvalue smaller faster consecutive transition shrinks corresponding components retaining algorithm gibbs sampling many important mcmc algorithms interpreted special case similar importance sampling uses proposal distribution case proposal distribution help design transition kernel sampling proposal accepted probability min proposed sample accepted kernel resulting procedure admits detailed balance condition stationary distribution important feature mcmc allows application graphical models possibility building valid transition kernels mixtures cycles transition kernels stationary distribution also stationary distribution cycle mixture cycling kernels gives gibbs sampling graphical models kernels markov blanket node also possible use block graphical models highly correlated variables blocked together mixing chapter approximate inference properties improve fact gibbs sampler method proposal distribution results acceptance probability similar general gibbs sampling fail kernel mix properly could happen variables strongly correlated principle one assemble neighbouring variables blocks update one however difficult regimes number variables flipped move one local optima another order total number variables makes approach intractable mixture kernels used combine global proposal local proposal fact could view message passing operator transition kernel least message passing exact mixture kernels gibbs sampling could produce interesting hybrid methods section combining gibbs sampling operator rephrased message update introduce new hybrid method hybrid methods stochastic methods slow convergence guaranteed converge even kernel reducible samples cover subset true support mcmc still converges single gibbs measure unique hand deterministic approximations fast difficult regimes modifications result convergence either generally intractable slow loop corrections methods tighten bound free energy degrade quality solutions moreover sampling methods flexible representing distributions motivated growing interest nonparametric approach variational inference particular variations belief propagation however sense methods rely markov chain inference closer variational inference mcmc methods better appreciate distinction consider two closely related methods gibbs sampling uses following update equation interestingly detailed balance condition equation gibbs sampler gives equation however given enough iterations gibbs sampling much accurate mean messages particles field method difference gibbs sampler equation enforced chain rather explicit averaging distributions means correlation information better taken account perturbed belief propagation consider single particle gibbs sampling updated according establish correspondence particle gibbs sampling set defining messages leaving variable factor messages def def gibbs sampling operator defines message function messages markov blanket adjacent factors completely define random operator note sample conditional distribution gibbs sampling combining operators lets write updates semiring substituting messages equation messages marginals equations get similar equation denotes message update operator distinction arguments also messages rather messages rewriting updates marginals equation identical form gibbs sampling distribution equation similar form allows combine operators linearly chapter approximate inference get perturbed operator def perturbed operator updates message calculating outgoing message according operators linearly combines get final massage iterations perturbed parameter gradually linearly changed towards algorithm summarizes procedure note updates perturbed compatible variable synchronous message update see section input factor graph number iterations output sample initialize messages repeat variable calculate using equation calculate messages using equation sample combine gibbs sampling messages end iterations return algorithm perturbed belief propagation experiments perturbed successful solving csps see chapter however also use marginalization sampling equation use ising model grids random graphs edges sampled local fields independently interactions change control problem difficulty higher values correspond difficult inference problems compared average logarithm perturbed updates also used fixed messages particles base mean variables marginal error log ising grid random graph figure log mean marginal error axes comparison perturbed gibbs sampling left periodic ising grid right random graph variables edges interactions size circle proportional difficulty problem figure compares perturbed gibbs sampling larger circles correspond difficult instances methods given maximum iterations perturbed gibbs sampling use iterations obtain sample ran convergence maximum number iterations reached results shows perturbed sampling method generally better gibbs sampling also cases result close random log marginal error perturbed produces relatively better results note difficult instances increasing even folds significantly improve results either gibbs sampling perturbed gibbs sampling explained formation pure states result exponential mixing time perturbed survey propagation operators sum prod apply perturbation scheme similar perturbed recall defines distribution fixed point therefore sampling distribution amounts randomly selecting single fixed point corresponds sampling single message bias message towards random choice recall marginal message identical corresponding message chapter approximate inference analogous equation algorithm alternative form perturbation perturb messages using implicit marginals recall using counting marginals marginals see equation simply frequency observing particular marginal fixed points implicitly defines marginal original domains denote obtaining sample bias outgoing messages accordingly similar perturbed gradually increased iterations perturbed use form perturbation section obtain satisfying assignment csps show although computationally expensive perturbed method often outperforms methods solving random csps part combinatorial problems message passing algorithms different semirings able solve variety combinatorial problems solve constraint satisfaction problems csps message passing often used defines uniform distribution solutions objective produce single assignment estimates partition function either using approximation given bethe free energy section decomposition integral section used approximate counting number solutions estimate partition function also used integration problems approximating permanent matrix see chapter iii semiring often used constrained optimization formulate bottleneck problems inference part thesis studies message passing solutions combinatorial problems three broad categories chapter studies constraint satisfaction problems use perturbed message passing section produce results solving random instances satisfiability coloring problems section chapter studies several problems including independent set packing construction codes chapter studies variations clustering problems including hierarchical clustering modularity optimization chapter study problems involve enumeration constraint satisfaction constrained optimization permutations includes bottleneck travelling salesman problem matching graph alignment graph isomorphism finding symmetries note classification combinatorial problems three categories superficial made solely provide organization several places violate categorization favour better flow example study constraint satisfaction problems sub isomorphism hamiltonian cycle chapter rather chapter investigate optimization counterpart csps chapter review message passing solutions finding trees rather clusters chapter moreover many graphical models presented proposed researchers included completeness final remark note many statements following assuming chapter constraint satisfaction saw section semiring formulate constraint satisfaction problems csps particular saw section several semirings isomorphic false true semiring therefore result equivalent procedures message update semiring called warning propagation marginals indicate whether particular assignment variable allowed therefore indicate cluster solutions however success warning propagation highly depends initialization messages contrast convergent fixed points semiring less dependent initialization example given graph problem asks whether possible assign one color node two adjacent nodes color variable constraints constraint depends two variables satisfied iff two variables different values identity function depends semiring see section figure set possible assignments variables solutions problem equation white circles corresponding problem equation factor prohibits single assignment chapter constraint satisfaction example given conjunction disjunctions literals seeks assignment evaluates true variables binary true false clause factor depends variables clause evaluates zero single assignment possible assignment variables consider following semiring formulation problem variables clauses factor corresponding first clause takes value semiring corresponds true except true true false case equal false figure shows set solutions true true true false false false false false true using semiring equation defines uniform distribution set solutions partition function counts number solutions challenge sample distribution estimate common approach sample use decimation see equation one repeatedly applies estimate marginals one fixes subset variables according marginals one may sample select maximum marginal applied reduced replaced argxi max process called repeated obtain complete joint assignment however using solving difficult problem marginalization fact equation showed using decimation one may estimate partition function problem suggests decimation may efficient approach solving npcomplete csps instead consider using perturbed belief propagation section sample set solutions semiring used perturbed better understand warning propagation perturbed applied csps consider following examples example apply three different message passing methods solve simple example figure warning propagation use semiring max prod version warning propagation example figure suggests set solutions clusters two subsets true true true false false false false false true clusters fixed point cluster two solutions corresponds following fixed point true true false false true true false false true true false false messages indicate allowed assignments within particular cluster solutions depending initialization messages may converge fixed points also include trivial cluster alternatively none assignments allowed applying problem starting uniform messages takes iterations converge maximum change marginals message similarly message opposite direction gives following approximate marginals true true true set solutions know correct marginals true true true error caused influential loops figure error rather small arbitrarily large instances sometimes prevent converging fixing value false sat problem equation collapses sat false chapter constraint satisfaction applies reduced problem give true note true true fixing false another round decimation yields solution false false true iii perturbed belief propagation perturbed find solution iterations see algorithm implementation shuffles order updates variables iteration first iteration means updates second iteration order updates end iteration true perturbed samples false marginal sample influences outgoing message according perturbed update equation turn influences beliefs end iteration true true final iteration order updates point true sample false means outgoing message deterministic false true choice propagates select false finally true true correctly shows choices produce solution compare performance perturbed general csps considered csp instances xcsp repository include global constraints complex domains instances intensive constraints functional form converted extensive format explicit representation using dense factors removed instances containing constraints enteries tabular form also discarded instances collectively enteries dense tabular form figure compares time iterations perturbed successful attempts methods satisfied instance overall perturbed solved instances successful successful runs hand average number iterations successful instances compared iterations perturbed makes perturbed times efficient since implementation represents factors dense tabular form remove many instances large factor size anticipate perturbed could probably solve many instances using sparse representation factors used convergence threshold terminated threshold reached iterations perform decimation sort variables according bias fix fraction biased variables iteration decimation fraction initially set divided time failed instance repeatedly applied using reduced times unless solution reached final attempt perturbed starting attempt increased factor case failure repeated times means perturbed used final attempt note perturbed uses number iterations maximum iterations per single iteration decimation also ran benchmarks maximum number iterations set phase transitions random csps perturbed time perturbed iters iters time figure comparison number iterations left time right used perturbed benchmark instances methods found satisfying assignments phase transitions random csps random csp rcsp instances extensively used order study properties combinatorial problems well analysis design algorithms studies rcsp critical phenomena focus geometry solution space function problem difficulty rigorous analyses confirmed geometric picture working large random instances scalar associated problem instance control parameter clause variable ratio characterize instance difficulty larger control parameter corresponds difficult instance many situations characterizes sharp transition satisfiability unsatisfiability example random random instance variables constraints generated selecting variables random constraint constraint set zero unsatisfied single random assignment control parameter example random control parameter random instances variables constraints average degree consider random graphs generate random instance sequentially selecting two distinct variables random generate edges large equivalent selecting possible factor fixed probability means nodes poisson degree distribution iterations reduced number satisfied instances also reduced average number iterations respectively still several folds expensive perturbed see appendix xyz details results chapter constraint satisfaction tight bounds problems finding exact location transition different csps still open problem besides transition unsatisfiability analyses revealed several phase transitions figure shows geometry set solutions changes increasing control parameter enumerate various phases problem increasing values control parameter replica symmetric phase symmetries set solutions ground states reflect trivial symmetries problem wrt variable domains example set solutions symmetric wrt swapping red blue assignment regime set solutions form giant cluster set neighboring solutions two solutions considered neighbors hamming distance one number variables local search methods often efficiently solve random csps belong phase figure schematic view set solutions csp varies increase control parameter left replica symmetric phase middle clustering phase right condensation phase small circles represent solutions bigger circles represent clusters solutions note view simplistic many ways total number solutions size clusters generally decrease left right clustering dynamical transition set solutions decomposes exponential number distant clusters two clusters distant hamming distance respective members divergent linear number variables condensation phase transition set solutions condenses dominant clusters dominant clusters roughly number solutions collectively contain almost solutions used even within condensation phase usually fails converge regime however cluster solutions clustering condensation phase valid rigidity transition included figure identifies phase finite portion variables fixed within dominant clusters transition triggers exponential decrease total number solutions leads unsatisfiability rough order dynamical rsb symmetry breaking general term indicating phenomenon system breaking symmetry governs behaviour selecting particular branch term replica symmetry breaking rsb originates technique replica trick first used analyze setting according rsb trivial symmetries problem characterize clusters solution order static rsb problems rigidity transition occurs condensation transition revisiting survey propagation figure schematic view demonstrates clustering condensation phase assume axes correspond considering whole space assignments highly correlated formation correlation distant variables breaks assume perturbed messages focused largest shaded ellipse case correlation significantly reduced picture summarizes first order replica symmetry breaking basic assumptions pitfalls decimation previously gave argument decimation based complexity marginalization integration recent analyses draw similarly negative conclusions effect decimation general picture point decimation process variables form correlations fixing one variable may imply assignment portion variables form loop potentially leading contradictions alternatively correlations result lack convergence error marginals may lead unsatisfying assignments perturbed avoids pitfalls two ways since many configurations probability final iteration perturbed avoid contradictions adapting recent choices contrast decimation variables fixed unable change afterwards backtracking schemes attempt fix problem decimation speculate simultaneous bias messages towards prevents formation correlations variables breaks see figure revisiting survey propagation studied random hyper graphs representing csps thermodynamic limit large random graphs locally means length short loops typically order log ensures absence correlations asymptotically exact set messages incoming node factor almost independent although messages remain uncorrelated condensation transition equations completely characterize set solutions clustering transition inadequacy indicated chapter constraint satisfaction existence set several valid fixed points rather unique instance better intuition consider cartoons figure middle right clustering phase middle corresponding axes highly correlated become correlated condensation right correlation variables far apart results correlation messages implies even loops long remain influential violates assumption messages uncorrelated results failure regime survey propagation comes picture solving csps going back algebraic notation using counting warning propagation semiring max prod initial semiring sum prod semiring computationally tractable initial semiring finite therefore message finite number values means message distribution possibilities however since indicates unfeasible case assignment allowed explicitly ignore message updates gives following update equations marginals applied csps example consider message figure summation equation possible combinations messages since messages assume one three valid values particular assignment total possible combinations enumerated summations equation however combinations form valid message update contribution calculating flavours decimation marginals equation also implies distribution original domain see equation similar use either implicit marginals revisiting survey propagation tion marginals equation perform decimation former case call select argxi max decimation later case call clamp argbpi max means outgoing messages variable node clamped way first case expect single assignment end decimation obtain cluster solutions subset assignments allowed however decimation process usually fixing subset variables marginals become close uniform indicating clusters solution preference particular assignment remaining variables happens apply random instances phase figure left point paramagnetic phase solutions form giant cluster local search method often efficiently find assignment variables yet fixed decimation original decimation procedure corresponds csp boolean variables slightly different choose fix cluster addition options corresponding respectively available however larger domains clear advantage example may choose fix variable first color allowed choose significant difference also reflected comparative see section computational complexity computational complexity update equation particular value needs consider every combination incoming messages take values minus empty set similarly using naive approach cost update equation however considering incoming messages one time perform exact update comparison cost updates two types message update see section see updates substantially expensive large domains higher order factors large perturbed survey propagation csp similar use perturbed max prod first semiring sum prod second semiring since perturbed seeks single assignment rather cluster solutions find satisfying solutions paramagnetic instances contrast paramagnetic cases returns trivial fixed point assignment allowed means opposed mostly applied random csps clustering condensation phase previous applications used heuristic decimation similar chapter constraint satisfaction perturbed used solve also random instances phase demonstrate applied perturbed benchmark csp instances figure maximum number elements factor less perturbed solved instances cases comparison perturbed solved instances making perturbed slightly better also solving problems satisfiability coloring examples introduced random procedures often used produce instances problems report results used procedures produce random instances variables control parameter report probability finding satisfying assignment different methods portion instances satisfied figure first row visualizes success rate different methods right left figure second row reports number variables fixed calling local search third row shows average amount time used find satisfying solution include failed attempts variations time includes time used local search final row figure shows number iterations used method level difficulty successful instances note include iterations local search variations area disk proportional frequency satisfied instances particular number iterations control parameter inference make following observations perturbed much effective remaining ten hundreds time efficient control parameter grows larger chance requiring iterations satisfy instance increases methods iii although computationally inefficient able find solutions instances larger control parameter suggested previous results many instances use iterations variables number iterations settings perturbed identical ones used compare perturbed coloring instances help decimation break initial symmetry problem fixing single variable arbitrary value use convergence threshold fix variables per iteration decimation perturbed perturbed use iterations methods use maximum iterations per iteration decimation methods failed find solution first attempt increased factor times final attempt avoid maximum iteration first iteration decimation increased similar setting variations see section decimation step maxi marginals close uniform consider instance run simplified instance number iterations rounded closest power two satisfiability coloring perturbed perturbed satisfiability rigidity satisfiability satisfiability satisfiability iters successful rigidity satisfiability time successful sec perturbed perturbed satisfiability satisfiability avg nontrivial fixes rigidity satisfiability success rate perturbed perturbed avg node degree clause variable ratio perturbed perturbed figure first row different methods various control parameters second row average number variables fixed using calling local search averaged instances third row average amount time seconds used successful setting method find satisfying solution includes time used local search forth row number iterations used different methods different control parameters method successful finding solution number iterations random instances rounded closest power include iterations used local search chapter constraint satisfaction fixed trivial cluster assignments allowed particularly pronounced instances fixes zero success rate solely due local search similar performance significantly outperforms table reports well average total iterations successful attempts method number iterations sum iterations used method plus iterations following observe perturbed solve easier instances using iterations see perturb result results also show difficult instances require method approximately correspond control parameter half instances satisfied larger control parameters usually result early failure satisfiability table suggests speculated section general preferable spdec particular applied coloring problem important advantage perturbed perturbed applied instances large factor cardinality variable domains example cardinality message makes perturbed impractical even able solve single instance around dynamical transition low perturbed satisfies instances problem graph partitioning cliques proved reduction complement nodes clique allowed color relation extends kcoloring factors ensure connected nodes different colors factors ensure nodes connected belong clique represents clique node factors inverse potts factors allow efficient calculation see section using denote set edges adjacent node following claim states complexity updates note condensation transition happens rigidity transition able find solutions rigidity would implied condensation transition marks onset difficulty however occur similar cases perturbed failed rigidity transition problem table comparison different methods method average number iterations including local search successful attempts reported approximate location phase transitions success rate avg iters success rate avg iters success rate avg iters success rate avg iters avg iters perturbed dynamical condensation transition satisfiability transition dynamical transition condensation transition rigidity transition satisfiability transition dynamical condensation transition rigidity transition satisfiability transition dynamical transition perturbed success rate ctrl param problem dynamical transition rigidity transition satisfiability transition dynamical transition rigidity transition condensation transition satisfiability transition chapter constraint satisfaction claim iteration message update factorp graph asynchronous message update proof complexity calculating message since factors one edge total cost messages becomes time complexity message markov blanket set nodes adjacent using update total cost messages becomes means cost messages update order however using asynchronous update node calculate messages since total cost dominates cost update experimental results within context scheme reduction formulation dominating set set cover set graph subset nodes size node adjacent least one member dominating set problem simple reductions set cover problem see formulations problems also closely related figure left induced problem solution right representation problem leader factors grey squares consistency factors black factor white dominating set set cover given universe set set subsets say covers iff member present least one member consider natural problem induced directed graph node define subset given set nodes connected let denote subsets induced set size covers equivalently induced subset vertices every node connected least one node example figure left induced solution indicated grey nodes consider undirected graph directed graph edges directions kdominating set equivalent induced problem moreover given problem instance construct directed graph induced equivalent given problem let collection nodes plus one node per subset define directed edges connect every representative moreover connect representatives directions easy show induced directed graph defines complexity problems one variable per edge note induced problem directed graph undirected difference affects representation two problems indicates node node associated node three types constraint factors ensure assignments define valid solution induced leader factors ensure node associated least one node admissible let set edges leaving node plus leader factor associated node consistency factors ensure node selected leader node node also selects leader alternative form factor factor allows efficient chapter constraint satisfaction variable update factor ensures nodes selected leaders figure shows example induced problem corresponding section saw possible calculate messages leader factors factor cost consistency factors pairwise alternative formulation claim message passing depending update schedule see section update asynchronous update proof assume consistency factors higher order form equation variable connected three factors therefore cost messages calculate messages simultaneously cost leader giving total adding cost factor total cost per iteration hand update separately previous costs multiplied gives clique problem independent set sphere packing given graph problem asks whether contains clique size least kclique problem closely related given graph set problem asks whether contains subset size least connection nodes relation problem analogous connection problems problem equivalent problem complement turn equivalent cover problem vertex cover subset nodes edge adjacent least one vertex easy see independent set iff vertex cover therefore solution independent set directly extends vertex cover clique problem independent set sphere packing special case isomorphism asks whether graph packing main graph complete graph size see section problems also closely related sphere packing finding nonlinear codes better motivate problems start problem formulated problem several show reduction problems simultaneously introduce message passing solutions three problems given symmetric distance matrix number codewords problem choose subset points minimum distance two maximized introduce two different inference obtains solution order establish relation problem csps need notion graph definition graph distance matrix defined graph binary variable let binary variables indicate subset variables size selected define factors assignment argx min max solution define following two types factors factor section ensures selected recall definition depends semiring equation problem defined semiring however since plan solve inference reductions important note reduction defined ring pairwise factors effective min max tabular form simply chapter constraint satisfaction use convert initial objective recall defines graph based distance matrix two nodes connected iff distance larger means corresponds graph connected nodes distance least proposition distance matrix defines uniform distribution cliques size proof since uniform domain enough show every clique size corresponds unique assignment given clique size define ident easy show need show constraint factors satisfied factor trivially satisfied pairwise factor equation becomes replaced min max operators semiring thresholded see pairwise constraint factors satisfied consider two cases nodes therefore also means connected definition implies therefore second term factor evaluates one either therefore first term evaluates one since pairwise factors factors every assignment corresponds unique clique size equation implies hand means factor satisfied therefore exactly variables nonzero therefore index variables identifies subset nodes connected forming clique claim iteration variable factor synchronous update message update completely asynchronous update original objective aims maximize minimum distance two clique problem independent set sphere packing figure using message passing choose random points euclidean plane maximize minimum pairwise distance iterations pbp touching circles show minimum distance proof cost calculation messages factor cost messages pairwise factors since factor cost evaluates since node adjacent nodes variablep synchronous update messages gives total asynchronous update cost messages factor need calculate message separately moreover updating message resulting total since updates asynchronous cost subsumes cost corollary complexity finding approximate solution reduction log using synchronous update figure shows example solution found message passing euclidean distance categorical variable define follows let set variables every two distinct points define factor max axi variable represents factor ensures distinct moreover distinct distance nodes chapter constraint satisfaction represent tabular form factor following proposition relates problem proposition distance matrix defines uniform distribution cliques size proof since defines uniform distribution support enough show clique size defines unique set assignments nonzero probability assignment defines unique clique size least first note basic difference former nodes connected distance later nodes distance least connected consider pairwise factor equation max basically replaced max semiring operator semiring thresholded axi clique size defines unique assignments assignment clique size define permutation nodes clique since permutations may define many assignments consider one assignment every two nodes since belong clique connected axi means pairwise factors defined equation values therefore assignment corresponds unique clique size let since pairwise factors defined equation therefore means connected forming clique size clique problem independent set sphere packing put simply acquiring set values table form zero value less set one otherwise resulting defines distribution means defines clique size graph connects nodes distance larger claim iteration pairwise factors synchronized update asynchronous update using reduction inference suggests log sync update log asynchronous update procedure problem proof claim since factors sparse complexity calculating single message resulting cost per iteration update however update message separately since message update costs total cost per iteration since diversity pairwise distances elements general cost finding approximate solution message passing log sync message update log async update second formulation first proposed find binary codes authors consider hamming distance binary vectors length obtain binary codes known minimum distance saw method grows exponentially number bits following section introduce formulation specific categorical variables hamming distance whose message passing complexity polynomial log using formulation able find optimal binary ternary codes large efficient sphere packing hamming distance defines distribution binary vectors length distance every pair binary vectors least better relate problem consider binary length nodes graph connect two nodes iff hamming distance least finding graph equivalent discovery nonlinear binary codes fundamental problem information theory see assuming odd number digits corrupted communication since every pair codes least digits apart still recover uncorrupted following collection ternary length obtained using discuss section every pair different least digits convenience restrict construction case binary vectors similar procedure may used find maximally distanced ternary vectors arbitrary chapter constraint satisfaction figure hamming distances distance factors white squares black construction involved previous constructions basic idea avoid exponential blow one variable per digit rather one variable per pair define auxiliary binary vector length indicates two different digit finally define constraint set auxiliary vectors ensures every two pair least different digits figure shows specifically let set binary vectors represents ith binary vector additionally two define auxiliary binary vector length distinct pair binary vectors particular digit auxiliary variable constrained iff define factor every pair ensure differ least digits clique problem independent set sphere packing factors defined follows every define factor depends three binary variables therefore explicitly define tabular form containing possible inputs difference binary ternary codes general tabular form factor example ternary codes tabular factor array always binary define factor section claim iteration variable factor synchronous update update completely asynchronous update proof claim first consider complexity updates auxiliary variable connected three factors connected since variables update messages async update next consider two possibilities updates update cost subsumed minimum cost also distance factors cost update since distance factor update total async update adding cost different scenarios get timecomplexities stated claim following table reports optimal binary codes including codes large number bits recovered using used perturbed update iterations find assignment chapter constraint satisfaction optimization variations csps section briefly reviews optimization variations csps studied far optimization version satisfiability known maximum weighted sat clause weight objective maximize weighted sum satisfied clauses simply using factors positive weight clause attempt find solution note constraint factor anymore see definition alternative approaches using variations energetic survey propagation also used improve results less studied optimization variation satisfiability adversarial sat corresponds inference minimum coloring chromatic number minimum problem since optimal value kmax bounded access oracle decision version incomplete solver message passing use binary search find minimum log kmax time decision problem particular since chromatic number bounded maximum degree approximating chromatic number using binary search message passing gives log maxi maxi time procedure approach used minimum minimum maximum clique maximum independent set minimum vertex cover however optimization variations also allow efficient direct approach note variation minimization maximization variations problems use binary search example minimum see section solved using decision problem parameter interest binary search former case later case threshold distance defines connectivity see definition however often variations minimization maximization also allow direct message passing solutions minimum replace semiring optimization variations csps sum semiring drop factor instead local factor gives weight node inference seeks subset nodes form dominating set largest sum weights also note changing semiring identity functions change accordingly leader factor consistency constraints remain valid gives efficient synchronous procedure minimum minimum dominatingset problem resulting message passing solution indeed variation affinity propagation respectively see section idea applies maximum clique maximum independent set alternative fixing maximizing size clique independent set may associate node weight seek subset nodes form independent set clique maximum weight sanghavi study message passing solution problem relation particular show starting uniform messages converges finds solution relaxation review maximum independent set using inference let set binary variables one node means independent set local factors capture cost negative weight node equal zero otherwise min pairwise factors ensure either easy see using variable synchronous update message passing performed efficiently weigt zhou also see propose interesting approach minimum vertex cover using energetic survey propagation one binary variable per node pairwise factor ensure edges covered least one node cover using semiring resulting fixed points warning propagation reveal minimal necessarily optimal vertex covers authors suggest using survey propagation decimation find warning propagation fixed points lowest energy smallest size cover chapter clustering clustering set central problem machine learning however many interesting clustering objectives including problems consider section section present message passing solutions several clustering objectives including modularity optimization message passing also used within expectation maximization obtain best results learning stochastic block models hidden variable model clustering message passing solution generalization clustering shallow trees proposed researchers however completeness review sections expressing kcenters problems inference problems sections given symmetric matrix pairwise distances number clusters seeks partitioning clusters associated cluster center sum distances cluster center minimized problem however exists several approximation algorithms metric distances present binary variable slightly different version objective proposed frey dueck simplified form messages known affinity propagation instead fixing number clusters objective modified incorporate cluster cost become center cost added sum distances cluster centers used decide number clusters let distance matrix directed graph moreover let denote willingness node center cluster simple heuristic set value uniformly define set binary variables one per directed edge indicates whether node follows node center node follow center following factors define cost constraints affinity propagation leader factors ensure node selects exactly one cluster center set edges leaving node consistency factors defined equation ensure node selects node center cluster node also selects center point note used dominating set problem section addition except using different semiring following factors local factors take distances willingness become center account min effect equal otherwise see equation definition semiring note fact min case means extensions semirings need consider different use operator however direct application inference problematic another reason since number clusters enforced soon node becomes center cluster nodes become centers without increasing value section resolve issue enforcing number clusters use inference semiring solve corresponding problem known problem complexity variable factor synchronous message update leader consistency factor allows efficient calculation messages respectively moreover messages leaving node calculated simultaneously using update equation facility location problem closely related problem facility location problem matrix specifies pairwise distance two parts bipartite graph goal select facilities minimize sum distances customer associated facility uncapacitated version problem restriction solved special problem message passing solution variations problem chapter clustering discussed discuss facility location special case problem section hierarchical clustering adding dummy node connecting cluster centers node cost think clustering previous section finding minimum cost tree depth two dummy node root shallow trees generalize notion allowing levels hierarchy objective find tree maximum depth minimizes sum edges alternative approach hierarchical clustering based nested application affinity propagation discussed previously presented model affinity propagation however possible obtain identical message updates using categorical variables selects one neighbouring nodes center cluster node case ignore leader factors change consistency local factors accordingly idea building hierarchies allow node follow another node even followed third node dropping consistency factors also forbidden however creates risk forming loops trick used add auxiliary depth variable node depth dummy node zero depth factors ensure follows spanning steiner trees although finding trees clustering problem since one could use techniques used clustering shallow trees include section given graph penalty per edge prize per node steiner tree objective select connected maximum sum prizes since optimal always tree construction similar shallow trees finds solutions versions problem main difference shallow trees root node member several tricks used find best set root however since may smaller different dummy node introduced zero cost connection nodes means node part steiner tree alternatively introduce dummy node moreover set node penalties zero result depth limited spanning tree bayati show problem maximum depth large enough convergent find minimum spanning tree problem given symmetric matrix pairwise distances number clusters clustering seeks partitioning minimizes maximum distance pairs partition formulate problem inference problem let set variables means point belongs cluster potts factor min equal two nodes cluster otherwise easy see using inference factor graph solution equation defines clustering minimizes maximum distances recall graph definition distance matrix graph claim clustering identical section proof factor equation min recall defines pairwise factor two nodes whenever whenever define factor however means reduced constraint factor satisfied therefore need consider cases gives factor two nodes use reductions approximate solution see equation considering cost message passing gives log log cost log cost binary search set possible pairwise distance values chapter clustering figure compares performance clustering using message passing furthest point clustering fpc triangle inequality holds note message passing solutions superior even using euclidean distance figure clustering points varying numbers clusters point average random instances ratio value obtained reduction using perturbed find satisfying assignments divided value furthest point clustering fpc left clustering random points euclidean space red line lower bound optimal result based worst case guarantee fpc right using symmetric random distance matrix problem given pairwise distance matrix problem seeks partitioning nodes one center per partition maximum distance node center partition minimized problem known even euclidean distance matrices however kcenter approximation algorithms apply distance matrix satisfies triangle inequality method dyer frieze similar furthest point clustering extends weighted problem distance point points scaled weight general case asymmetric distances allow approximation however log exists define whose inference results optimal solution problem consider graph induced distance matrix let set variables indicates whether center cluster define following factors problem figure problem local factors black squares leader factors light grey consistency factors white factor dark grey local factors min leader consistency factors defined case induced set section need replace version equation variants problem capacitated additional constraints points group may added factors claim clustering identical section proof leader consistency factors identical factors used identical difference one variable per whereas one variable per considering local factors min see assume aij drop variables factorgraph also omit local factors effect gives section similar problem use reduction find solutions chapter clustering figure left clustering random points plane various numbers clusters ratio value obtained reduction perturbed value right location formulated asymmetric problem solved using message passing yellow squares indicate potential facility locations small blue circles indicate customers task select facilities red squares minimize maximum distance customer facility radius circles value problem binary search seeks optimal collective range factors since local factors take values search range basically values adds additional log multiplicative factor complexity depends message update significantly reduce number variables complexity bounding distance center cluster given upper bound may remove variables assuming nodes distance every node complexity inference synchronous update drops log upper bound obtained example applying approximation algorithms figure left compares performance triangle inequality holds facility location problem also formulated asymmetric problem distance customers distance facility another facility figure right modularity maximization widely used objective clustering community mining modularity maximization however exact optimization modularity modularity closely related fully connected potts graphical models many proposed various heuristics modularity optimization brief review potts model section introduce representation problem large number factors section use augmentation technique section incrementally incorporate violated constraints modularity maximization models potts model let undirected graph adjacency matrix let normalized adjacency matrix also def let denote normalized degree node graph clustering using modularity optimization seeks partitioning nodes unspecified number clusters maximizing def first term modularity proportional second term proportional expected number within cluster null model weighted node degrees node null model graph resolution parameter default set one influences size communities higher resolutions motivates larger number clusters potts model node associated variable kmax kmax upper bound number clusters pair variables pairwise interaction inference fully connected gives assignment node cluster maximize modularity clique model introduce alternative modularity optimization introducing representation suggest procedure stochastically approximate null model using sparse set interactions generate random sparse null model null weighted edges null randomly sampling two nodes drawn independently connecting weight proportional bnull already connected weight added current weight repeat process times however since edges repeated total number edges sparse null model may less finally chapter clustering normalized sparse null model def bnull null anull easy see generative process expectation produces fully connected null representation use following formulation let null set binary variables let denote cardinality null variable equal one means corresponding edge present final model goal define final model consists cliques define factors follows local factor variable equal difference weighted adjacency null model edge present null min enforcing formation cliques minimizing sum local factors negative sum local factors evaluates modularity equation three edges null form triangle define clique factor factors ensure formation cliques two edges adjacent node present third edge triangle also present computational challenge large number clique constraints fully connected null model need factors even using sparse null model assuming random edge probability graph triangles graph recall null brandes first introduced formulation similar form constraints however since include constraints beginning null model fully connected method applied small toy problems choice using square root weighted degrees sampling weighting reduce variance one may also use pure importance sampling use product weighted degrees sampling set null model uniformly uniform sampling edges null model set product weighted degrees modularity maximization simplified message update augmentation give technical details simplify message update clique factor satisfied either zero one three variables domain therefore order derive message updates clique factor variable particular value apply operator minimize valid cases incoming messages clique factor zero gives simplified messages min min minimization three feasible cases two feasible cases work normalized messages use denote applies marginal scalar called bias means shows preference normalizing messages get following form simplified messages clique constraints min min order deal large number factors use augmentation approach section start run obtain solution may even unfeasible find set clique constraints violated current solution augment factors enforce constraints order find violated constraints current solution simply look pairs positively fixed edges around node third edge triangle positively fixed add corresponding clique factor see algorithm appendix details modularity maximization experiments experimented set classic since optimization criteria modularity compared method best known modularity optimization heuristics fastmodularity note different standard normalization semiring minxi obtained form mark newman website http chapter clustering figure left clustering power network message passing different clusters different colors nodes scaled degree right clustering politician blogs network message passing liberal conservative table comparison different modularity optimization methods wkarate netscience dolphins lesmis celegansneural polblogs karate time modularity modularity time modularity time time modularity modularity time cost football time polbooks modularity louvian cost fastgreedy edges nodes weighted message passing sparse problem message passing full modularity maximization louvain leading eigenvector table summarizes results see also figure method report time seconds modularity communities found method table include results message passing full sparse null models used constant generate stochastic sparse null model message passing also included null saving cost using augmentation column shows percentage number constraints considered augmentation example cost polblogs shows augmentation sparse null model meant using times fewer compared full overall results suggest method comparable terms time quality clustering although note number triangle constraints large dense graphs increases quickly deteriorates performance approach despite using augmentation despite fact results confirm utility augmentation showing able find feasible solutions using small portion constraints message passing use median anull null tmax perform decimation directly fix variables based bias chapter permutations matching permanent integration maximization problems unrestricted permutations define several important combinatorial problems two notable examples integration problems permanent determinant matrix determinant matrix defined det sign set permutations elements symmetric group index ith element particular permutation sign classifies permutations even sign odd sign perform even odd permutation even odd number pairwise exchanges difference definition permanent removal sign function perm see permanent determinant closely related two easy combinatorial problems graphs perfect matching spanning calculating permanent determinant obtained theorem states number spanning trees graph adjacency matrix equal det arbitrary laplacian diagonal matrix degree node diagonal row column removed intermediate step representing permutation graphical model use bipartite matching permanent graph perfect matching one one mapping elements represented using corresponding edges easy see perfect matching identifies permutation maximum weighted matching assignment problem problem find perfect matching max bottleneck assignment problem seeks min maxn representation next section shows bipartite matching bottleneck assignment problems correspond inference computation permanent corresponds inference interestingly inference setting indeed application find maximum weighted matching generalization maximum weighted one cases loopy guaranteed optimal although mcmc methods section provide polynomial time approximation schemes permanent many combinatorial integration problems found slow practice motivated approximations using deterministic variational techniques particular guaranteed provide lower bound permanent complexity review bayati maximum bipartite matching given bipartite graph associated matrix entries define two sets variables mean node connected node obviously representation redundant pairwise factor ensure consistent consistency factors ensure consistency local factors represent cost matching connected matching two local factors account figure shows easy see joint form equal consistent assignment equal otherwise therefore chapter permutations figure matching local factors black consistency factors white squares maxx produce permanent matching respectively cost iteration moreover optimal solution unique guaranteed converge solution iterations difference cost first second best matchings cost best matching alternative use binary variable model edge bipartite graph associated binary variable replace consistency factors degree constraint ensure node matched exactly one node however model results updates equivalent one simplification updates discussed exactly updates binary variable model semiring maxn consistent assignment evaluates min therefore inference seeks assignment minimizes maximum matching cost bottleneck assignment problem arbitrary graphs also use message passing solve matching arbitrary graph adjacency matrix proposed related task counting perfect matchings arbitrary graph edge assigned one binary variable degree factors node restrict number values one traveling salesman problem set edges adjacent node sanghavi consider problem maximum weighted arbitrary graphs changing degree factor also local factor takes weights account show solution corresponding relaxation integral converges optimal solution moreover converge solution integral matchings arbitrary graph also related permanent also vertex disjoint cycle covers set directed cycles cover nodes exactly number cycle covers graph equal perm turn equal square number perfect matchings fact directed cycle cover maximum weight equivalent maximum weighted bipartite matching construction previous section section use message passing obtain minimum weighted undirected cycle cover restrict covers obtain minimum weighted cover single cycle minimum tour tsp traveling salesman problem traveling salesman problem tsp seeks minimum length tour cities visits city exactly tsp general distances constant factor approximation problem possible best known exact solver due held karp uses dynamic programming reduce cost enumerating orderings opment many standard optimization techniques closely linked advances solving tsp important examples simulated annealing mixed integer linear programming dynamic programming ant colony optimization genetic algorithms since dantzig manually applied cutting plane method problem combination sophisticated cuts used techniques produced concorde notable results large instances reported heuristic continuously improves solution exchanging nodes tour readable historical background tsp applications see search optimal tour search permutations cities contains constrained dot return starting city without visiting cities producing permutation minimum cost may include called vertex disjoint see section provide two approaches model tsp section presents first approach ignores chapter permutations subtour constraints finds cycle covers augment constraints become violated augmentation process repeated feasible solution found second approach presented section use variables represent node visited number cities subtour constraint automatically enforced second formulation computationally expensive closely related isomorphism see section one think problem finding hamiltonian cycle finding isomorphic loop size augmentative approach let denote graph problem positively weighted symmetric adjacency matrix objective select subset identifies shortest tour cities let set binary variables one edge graph means tour use refer variable recall node denotes edges adjacent define factors follows local factors represent cost associated edge min either zero valid tour satisfies following necessary sufficient constraints constraints degree factors ensure exactly two edges adjacent vertex tour subtour factors ensure loops contain strict def subsets nodes enforce define set edges one end end need least two edges leaving subset following set factors enforce constraints three types factors define whose minimum energy configuration smallest tour tsp therefore use inference obtain optimal tour note traveling salesman problem subtour degree constraints depend large number variables however due sparsity allow efficient linear time calculation messages see equation significant computational challenge complete tsp subtour factors one subset variables figure address problem using augmentation simplified messages section introduced factors inference degree subtour factors different variations types factor simplicity work normalized message equivalent assuming notation used message marginal belief refer normalized marginal belief bias recall degree constraint node depends variables edges adjacent review message min show message update simplifies min min order satisfy degree constraint one hand know messages normalized means ignored summation equation order satisfy constraint factor two adjacent variables value therefore seek two incoming messages minimum values let mink denote smallest value set min combine updates get normalized message simply negative second largest incoming message excluding degree factor min following similar procedure messages subtour factors given max min searching minimum incoming message encounter two messages chapter permutations negative zero values safely assume stop search results significant speedup practice note equation equation need calculate second smallest incoming message corresponding factors less current outgoing message asynchronous calculation messages minimization repeated outgoing message however update finding three smallest incoming messages factor calculate messages time iteration iteration iteration iteration figure message passing results augmentation step complete graph printing board instance blue lines figure show selected edges end message passing pale red lines show edges bias although negative close zero augmentation deal exponentially large number subtour factors use augmentation procedure section starting subtour factor find solution using solution feasible subtours done otherwise find subtours finding connected components identify variables subtour add subtour factor ensure constraint satisfied next iteration augmentation speed message passing reuse messages previous augmentation step moreover obtaining marginals ensure degree constraint violated node two neighbouring edges figure shows iterations augmentation print board tsp instance algorithm appendix gives details message passing solution tsp also uses several minor tricks speed message updates traveling salesman problem nodes nodes augmentation iters nodes subtour factors nodes nodes augmentation iters nodes nodes random matrix optimality nodes nodes nodes nodes correlation distance nodes optimality time sec nodes hamming distance augmentation iters nodes nodes optimality time sec augmentation iters time sec nodes subtour factors optimality random euclidean distance subtour factors nodes subtour factors subtour factors time sec nodes augmentation iters greedy tsplib instances optimality time sec nodes nodes figure results message passing tsp different benchmark problems left right plots show running time optimality ratio compared concorde iterations augmentation number subtours constraints function number nodes optimality relative result reported concorde note plots except optimality plots linear trend shows monomial relation axm values axis slope shows power chapter permutations experiments evaluate method five benchmark tsplib contains variety benchmark instances majority euclidean geographic euclidean distance random points iii random symmetric distance matrices hamming distance random binary vectors fixed length bits appears applications data compression radiation hybrid mapping genomics correlation distance random vectors random features using tsp gene producing random points features well random distances iii used uniform distribution cases report optimality number iterations augmentation number subtour factors final iteration experiments use concorde default settings obtain optimal results figure column left reports optimality ratio ratio tour found message passing optimal tour demonstrate instance also report optimality ratio two heuristics optimality guarantees metric instances nearest neighbour heuristic incrementally adds end current path closest city form loop greedy algorithm log incrementally adds lowest cost edge current avoiding subtours plots figure except second column format using plot linear trend shows monomial relation axes axm indicates slope line plot intercept corresponds log studying slope linear trend left column figure observe almost instances message passing seems grow slope exceptions tsplib instances seem pose greater challenge random distance matrices seem easier message passing similar trend suggested number subtour factors iterations augmentation slope suggesting linear dependence exceptions tsplib instances grow faster random distance matrices seem grow overall observe augmentative able find solutions polynomial time although powerful methods concorde able exactly solve instances several thousands variables general random benchmark instances experiments used full graph means iteration message passing number subtour factors experiments use tmax iterations median damping used decimation fixed remaining variables per iteration decimation note fixing top variables positive bias remaining variables automatically clamped zero increases cost message passing log multiplicative factor however often produces better results geographic distance distance surface earth large sphere many larger instances concorde default setting using cplex solver able find optimal solution nevertheless used upper bound optimal produced concord evaluating method traveling salesman problem remains exponential approximation random instances appears using pairwise factors present alternative finding permutations without subtours formulation factors therefore complete remains tractable however practice inference effective augmentation approach therefore use solve version tsp known bottleneck tsp given asymmetric distance matrix task bottleneck traveling salesman problem btsp find tour points maximum distance two consecutive cities tour minimized approximation arbitrary instances problem let denote set variables represents node visited assume modular arithmetic module members mod mod pair variables define factor min max max tabular form diagonal enteries ensures moreover means cities visited consecutively factor effect however two cities visited one depending whether visited represent distance factor easily converted domains replacing formulation similar symantics proposed however authors applied message passing solution instance five cities chapter permutations identity values tabular form alternatively replace identity min max operations equation corresponding functions relate uniform distribution hamiltonian cycles use reduction find solutions bottleneck tsp recall graph connection node iff proposition distance matrix btsp defines uniform distribution directed hamiltonian cycles proof first note defines uniform distribution support unnormalized value zero one distinguish two hamiltonian cycles different starting point otherwise represent tour consider pairwise factor equation every hamiltonian cycle defines unique assignments given hamiltonian cycle ith node path define show pairwise factors equation consider two variables consecutive hamiltonian cycle assume consecutive appears means therefore turn means since pairwise factors every defines unique hamiltonian path given assignment construct show defines hamiltonian path every two variables one indicator functions equation evaluate one means first implies since values distinct two variables recall using modular arithmetic pairwise factor equation means therefore definition consecutive nodes therefore hamiltonian path proposition implies use reduction solve hamiltonian cycle problems resulting hamiltonian cycle problem special case technique adjacency matrix one graph directly used build factors second graph tabular form factor suggests second graph simple loop length see section traveling salesman problem figure solution using reduction bottleneck tsp different number cities euclidean space left well asymmetric random distance matrices right perturbed figures show one standard deviation random instances expected due sparse form pairwise factor perform updates efficiently claim messages reduction factor equation obtained proof factors equation given factor depending order takes several forms assuming messages normalized message given therefore cost calculating message normalizing message since pairwise factors gives iteration sumproduct solving hamiltonian cycle problem reduction log log factor cost binary search see equation chapter permutations figure reports average performance message passing instances well lower bound optimal value tours different length report results random points euclidean space well asymmetric random distance matrices euclidean problems lower bound maximum distance two closest neighbors node asymmetric random distance matrices maximum minimum length incoming edges minimum length outgoing edges graph matching problems consider two graphs weighted adjacency matrices respectively enumerate several important problems permutations based two graphs section introduces graphical model problems isomorphism monomorphism study message passing solution graph homomorphism use find symmetries graphs section introduce general graph alignment based previous work bradde show model problems quadratic assignment problem maximum common subgraph common idea settings variation adjacency matrix graph used pairwise factor edges graph adjacency defining markov network pairwise factors isomorphism graph isomorphism problem asks whether identical permutation nodes written seeks mapping abuse notation also write although polynomial time solutions special instances graph isomorphism problem general case remained elusive much know whether problem see permutation recall symmetric group called automorphism automorphisms composition form group called automorphism group aut automorphism group also defines natural notion symmetry nodes graphs orbit node set nodes mapped automorphism def orbit aut orbits partition set nodes group nodes sense symmetric makes prime candidate defining symmetry complex networks one node minimum length edges point city second minimum length edges also considered calculating tighter bound graph matching problems isomorphism asks whether isomorphic vertex induced subgraph seeks injective mapping dealing morphisms assume mapping smaller graph larger graph therefore subgraph isomorphism defined follows one variable per domain variable two types pairwise factors edge factors ensure edge mapped edge assuming tabular form factor semiring simply adjacency matrix factors ensure mapped semiring tabular form takes simple form binary valued adjacency matrix using semiring fully connected markov network defines uniform distribution subgraph isomorphisms could use sumproduct decimation perturbed sample individual assignments assignment injective mapping means node mapped node particular integral equal cardinality automorphism group two nodes orbit iff marginals orbit also suggests procedure finding approximate symmetries graphs use find marginals group nodes based similarity marginals however cost message passing graphical model important barrier practice claim assuming using variable synchronous update time complexity iteration subgraph isomorphism proof first calculate cost sending messages edge factors edge factors row tabular form factor chapter permutations ply entries corresponding message add values column obtain outgoing message procedure depends number entries procedure similar factors therefore overall cost sending messages factors using variable synchronous update calculating messages takes using assumption claim overall cost therefore however possible improve complexity considering sparse mappings example restrict domain variable nodes degree node furthermore nodes also neighbours degree neighbours node subgraph monomorphism supermorphism monomorphism relaxes constraint subgraph isomorphism injective nodes mapped distinct nodes however allowed cover subset edges induced subgraph note previous graphical models introduce isomorphism fact define monomorphism difference isomorphism factors replaced following uniqueness factors uniqueness factors inverse potts factors ensure disconnected nodes mapped different nodes mapping injective despite difference interested automorphisms generally distribution defined monomorphism identical isomorphism claim assuming using variable synchronous update time complexity iteration monomorphism proof complexity sending messages edge factors see proof claim however uniqueness factors inverse potts factors allow bradde suggest trick reduce unfortunately details omitted unable follow route graph matching problems calculation messages means overall cost sending messages factors cost calculating messages using update also gives overall complexity per iteration far seen two variations mapping graph subgraph subgraph isomorphism image strictly agrees monomorphism image contained within third possibility ask mapping image contains injective due lack better name call mapping subgraph supermorphism supermorphism two types factors seen factors equation ensure image contains uniqueness factors edges ensure mapping injective claim assuming using variable synchronous update time complexity iteration subgraph supermorphism proof complexity sending messages factors see proof claim uniqueness factor require computation means overall cost sending messages factors cost calculating messages using update also gives overall complexity per iteration graph homomorphism graph homomorphism relaxes constraint monomorphism homomorphic mapping map two distinct nodes node however equation hold therefore two nodes adjacent still mapped distinct adjacent nodes compared areas graph theory study rich structure surprising properties homomorphisms relatively recent study graph homomorphisms covers diverse areas including property testing graph sequences limits constraint satisfaction problems many areas interested counting number homomorphisms graph see shortly graph homomorphism reduces many interesting csps therefore hard show moreover counting number chapter permutations figure left number endomorphisms distinct graphs nodes compared integral larger disks represent graphs smaller number nodes right comparison normalized marginals exact marginals endomorphism graphical model homomorphisms set homomorphisms graph endomorphisms composition form endomorphism monoid see definition monoid identity element simply maps element interest endomorphism conjecture use instead automorphism approximate symmetries graphs graphical model representation homomorphism investigated different contexts knowledge message passing previously used counting finding homomorphisms markov network homomorphism resembles isomorphism monomorphism variables contains equation assuming easy see complexity variablesynchronous message passing makes method efficient sparse graphs graphical model extended represent homomorphism weighted graphs define small graphs exactly count number homomorphisms endomorphisms evaluate accuracy message passing compared estimates exact number endomorphisms isomorphically distinct graphs instances figure reports result well accuracy marginals suggesting despite existence short loops graphs estimates relatively accurate graph matching problems reduction csps relation homomorphism csps review relation section csp encountered thesis coloring clique cover see example section corresponds finding homomorphism complete graph order similarly homomorphism complement corresponds relation also reflected corresponding adjacency matrix inverse potts model used clique problem see section recall problem corresponds finding clique size special case isomorphism case identical monomorphism homomorphism reduction model section fully connected markov network edge factors used isomorphism monomorphism homomorphism equation similarly set problem equivalent finding homomorphisms complement independent set alternative relation graph homomorphism homomorphism graph two connected nodes one nodes defines independent set hamiltonian cycle problem corresponds monomorphism cycle length alternatively formulate subgraph supermorphism reduction formulation bottleneck tsp section indeed supermorphism finding symmetries one may characterize graph using number homomorphism graphs characterization behind application graph homomorphism property testing definition graph sequences let hom set homomorphism set assignments let sequence graphs whose number nodes vector defined uniquely identifies isomorphism rather identifying particular graph within set graphs interested identifying node single graph within set nodes note identifications isomorphism objective equivalent finding orbits approach finding orbits using graph homomorphism rather isomorphism founded following conjecture chapter permutations figure table right shows unnormalized endomorphism marginals graph left row corresponds normalization gives number row column number times node mapped endomorphism total number endomorphisms graph orbits however node maps nodes frequency nodes however mappings node column table remains different mappings predicted conjecture nodes similar rows columns belong orbit conjecture given uniform distribution endomorphisms necessary sufficient condition orbit orbit orbit note simply relative frequency mapping node node endomorphism therefore conjecture simply states node equivalent automorphism necessary sufficient frequency mapping nodes trivial prove necessity found difficult prove statement direction therefore similar related conjectures turn computational verification experimented distinct graphs nodes instances obtained exact marginals constructed orbits suggested conjecture obtained exact orbits using software mckay piperno cases two partitioning nodes orbits identical also note conjecture restricting condition similar frequency mappings nodes sufficient figure shows weaker condition example figure left used homomorphism marginals find approximate symmetries graph visible symmetries known thomassen graph obtaining marginals used principle component analysis pca whitening extract three values graph matching problems figure coloring nodes reducing dimensionality marginals three dimensions rgb using pca whitening left marginals calculated using right marginals kronecker graph obtained using gibbs sampling annealing graph middle product two graphs top right two nodes connected iff corresponding nodes connected note product two colors produces color product graph similarly colored nodes similar neighbours rgb colors figure suggests approach able identify similar nodes similar colors dense graphs message passing longer accurate figure right use gibbs sampling annealing estimate endomorphism marginals kronecker product two random graphs use pca color nodes note algorithm unaware product format choice product graph easily observe fact product similarly colored nodes produce similarly colored nodes product graph alternative use spectral clustering matrix marginals obtain clusters related context krzakala use matrix find symmetric clusters stochastic block models note basic difference approach matrix used spectral clustering notable matrices used within context laplacian matrix modularity matrix bethe hessian relation clustering conventional sense orbits graph better understood extreme case clusters form isolated cliques identical orbits graph alignment graph alignment problem seen optimization counterpart decision problems isomprphism monomorphism homomorphism past different methods tried optimize chapter permutations variety different objectives context message passing two distinct approaches used bayati propose sparse graph alignment show scales large instances term sparsity refers number edges also restricted possibility matching nodes node match predetermined nodes used authors resembles binary version maximum bipartite matching section bradde used minimize number misalignment similar graph monomorphism follow route distinction suggest using semiring isomorphism account different matching costs using several tricks consider general objective function evaluates mapping null show optimize objective using inference node matching preference matching node edge matching preference matching node merging preference mapping nodes node node deletion preference penalty ignoring node mapping null node edge deletion preference preference dropping edge insertion preference preference adding edge matched edge define preferences way optimal solution also solution interesting decision problem example optimal alignments following parameters reproduce hom node matching edge matching node merging node deletion edge deletion edge insertion alternatively using positive uniform node edge matching preferences set merge cost allow node deletion zero cost optimal solution maximum common subgraph particular common subgraph use node matching edge matching graph matching problems node merging node deletion edge deletion edge insertion given two weighted adjacency matrices respectively flow facilities distance locations quadratic assignment problem seeks mapping facilities locations order optimize flow max assume weights positive set alignment preferences optimize quadratic assignment problem node matching edge matching node merging node deletion edge deletion edge insertion define factors based various alignment preferences one variable per node null null corresponds ignoring node mapping alignment three type factors local factors take node matching preferences node deletion preference account null otherwise edge factors defined edge partly account edge matching node merging edge deletion null null otherwise chapter permutations factors defined edge partly account node merging edge insertion null null otherwise general form fully connected cost factors log means iteration variable synchronous log however matching candidates limited sparse alignment graphs sparse cost significantly reduced practice example figure shows matching metabolic network distorted version edges removed number random edges added generated matching candidate node original graph including correct match randomly selected nodes used graph alignment following preferences match original graph distorted node matching edge matching node merging node deletion edge deletion edge insertion observed message passing using able correctly match nodes two graphs used initial iterations damping parameter used decimation fixed variables iterations total less minutes graph matching problems figure matching metabolic network highly distorted version using message passing edges original network left removed number random edges added produce distorted network node random candidates matching including correct choice message passing able identify correct matching accuracy conclusion thesis studied general form inference graphical models emphasis algebraic abstractions organized important subset inference problems inference hierarchy studied settings distributive law allows efficient approximations form message passing investigated different methods improve approximation loopy graphs using variational formulation loop correction survey propagation hybrid techniques studied graphical modelling combinatorial optimization problems different modes inference inference optimization framework graphical modeling pros cons cons using graphical models combinatorial optimization twofold implementing message passing procedures compared standard techniques using integer programming solvers complex time consuming complicated fact standard guideline designing representation minimize computational complexity increase quality message passing solution indeed used many tricks efficiently approximate solution problems example include simplification messages alternative normalization augmentation variable message update introduction auxiliary variables using damping decimation etc hand dealing large scale difficult optimization problems one resort conceptual computational decomposition graphical modelling message passing techniques immediate candidates message passing mass parallelizable scalable often finds solutions providing diverse set combinatorial problems thesis also attempt establish universality message passing course problems better lend graphical modelling others resulting better computational complexity quality results table summarizes important information message passing solutions combinatorial problems proposed reviewed thesis graph matching problems table summary solutions combinatorial problems time complexity one iteration message passing report different costs different update schedules problem see text references number nodes number problem belief propagation semiring ops min max perturbed survey propagation min max independent set async async reduces async async async min similar async binary var model min max log reduction min max min max max async categorical var model log reduction max min vertex cover reduces gibbs samp relation others async schedule complexity hamming digits min dist async min facility location min min span tree min steiner tree min affinity propagation min max log reduction min max log reduction modularity max min clique model using augmentation kmax max matching min potts model cycle cover bottleneck assignment min max max min tsp min bottleneck tsp min max log async reduction hamiltonian cycle subgraph isomorphism subgraph monomorphism subgraph supermorphism homomorphism graph alignment max log general costs max common variation graph alignment quadratic assignment variation graph alignment chapter permutations future work due breadth models problems covered thesis investigation lacks deserved depth many cases particularly pronounced experiments moreover encountered many new questions possibilities preparing thesis enumerate topics demand depth investigation future work many problems discussed also efficient relaxations times come approximation guarantees comprehensive experimental comparison message passing relaxations problems terms speed accuracy highly desirable algebraic approach inference suggests message passing procedures discussed including survey propagation also applicable domain complex numbers extensions domain may allow new applications using fourier coefficients factors may also produce better solutions many problems studied solving csps study using graph homomorphism application finding symmetries work progress particular relation methods stochastic block models spectral techniques needs investigation although preliminary results ising model suggested using reductions inference performs much better direct message passing extensive comparison two approaches inference missing analysis section noted several optimization counterparts csps allow using addition direct optimization approach using binary search expensive problems know approach performs better practice references achioptas sorkin optimal myopic algorithms random foundations computer science proceedings annual symposium pages ieee achlioptas algorithmic barriers phase transitions mar achlioptas naor peres rigorous location 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complex systems zhou wang xiao partition function expansion region graphs messagepassing equations journal statistical mechanics theory experiment appendix appendix references input graph normalized weighted adjacency maximum iterations tmax damping threshold output clustering nodes construct null model null true augmentation loop tmax loop null anull update beliefs calculate using equation end max update msgs end end end null add corresponding clique factor end end factor added break loop else end connectedcomponents null algorithm message passing modularity maximization input graph weighted symmetric adjacency matrix maximum iterations tmax damping threshold output subset edges tour construct initial initialize messages degree constraints initialize true augmentation loop tmax loop including factor updates outgoing messages find three lowest values calculate using equation max end end end respecting degree constraints connectedcomponents return else augment initialize end algorithm message passing tsp
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concentration behavior penalized least squares estimator oct penalized least squares behavior alan muro sara van geer muro geer seminar statistik eth abstract consider standard nonparametric regression model take estimator penalized least squares function article study closeness true function complexity penalization estimator complexity described seminorm class functions first present exponential concentration inequality revealing concentration behavior penalized least squares estimator around nonrandom quantity quantity depends problem consideration conditions proper choice tuning parameter obtain bounds nonrandom quantity illustrate results examples include smoothing splines estimator keywords concentration inequalities regularized least squares statistical introduction let independent response variables satisfying given covariates space unknown function given space independent standard gaussian random variables assume smooth sense degree smoothness unknown make clear estimate consider penalized least squares estimator given arg min tuning parameter given seminorm vector write apply notation vector function moreover avoid digressions main arguments assume expression exists unique function define expression seen description term measures closeness true function quantifies smoothness mentioned assume complex degree complexity known accounts assuming upper bound unknown therefore choose model class important see taking model class fixed instead could lead model misspecification error unknown smoothness large estimators roughness penalization widely studied wahba green silverman consider smoothing estimator correr splines sponds solution denotes derivative provides results general penalized likelihood estimator general quadratic functional complexity regularization assume functional seminorm noise follows gaussian distribution penalized likelihood estimator reduces estimator upper bounds estimation error found literature see van der vaart wellner del barrio complexity regularization term included standard method used derive roughly follows first basic inequality sup invoked inequality right hand side obtained functions finally upper bounds estimation error obtained high probability using entropy computations penalty term included similar approach used case process studied terms smoothness functions considered limitation approach mentioned allow obtain lower bounds consistency results proved van geer wegkamp clear use derive explicit bounds chatterjee proposes new approach estimate exact value error least square estimators convex constraints provides concentration result estimation error relating expected maxima gaussian process one use result get upper lower bounds van geer wainwright direct argument employed show error general class penalized least squares estimators concentrated around expectation penalty assumed convex moreover authors also consider approach chatterjee derive concentration result uniformly bounded function classes general loss functions goal paper contribute study behavior theoretical point view consider approach chatterjee extend ideas penalized least squares estimator penalty based squared seminorm without making assumptions function space present concentration inequality showing concentrated around nonrandom quantity defined nonparametric regime depends sample size problem consideration derive upper lower bounds theorem obtained proper choice two additional conditions including entropy assumption combining result theorem one obtain upper lower bounds high probability sufficiently large sample size illustrate results examples section observe able recover optimal rates convergence estimation error literature introduce notation used following sections denote minimum rmin min let minimum attained fmin note fmin seen unknown noiseless counterpart rmin nonrandom unknown quantity two vectors let denote usual inner product rmin define sup additionally write emn ehn moreover define random quantity arg max nonrandom quantity arg max lemma follow unique ease exposition use following asymptotic notation throughout paper two positive sequences write lim sup moreover employ notation addition make use stochastic order symbols important note quantities rmin depend sample size however omit dependence notation simplify exposition organization paper first section present main results theorems note former require assumptions section latter needs two extra conditions introduced stating theorem section illustrate theory examples section present concluding remarks finally section present proofs deferred appendix results literature used last section behavior main results first theorem provides concentration probability inequality around seen extension theorem chatterjee penalized least squares estimator squared seminorm penalty term theorem exp asymptotics satisfies therefore random fluctuations around negligible size comparison moreover asymptotic result implies asymptotic distribution degenerate theorem provide bounds additional conditions observe satisfies condition remark note consider square seminorm penalty term generalization results penalties form straightforward omitted simplicity case considered study since method proof requires square root penalty term convex observed proof lemma stating first condition required theorem introduce following definition let subset metric space number smallest value exist min moreover number largest value exist condition let rmin log log condition seen description richness refer reader kolmogorov tihomirov birman solomjak extensive study entropy bounds van der vaart wellner van geer application empirical process theory condition rmin condition relates roughness penalty minimum achieved functions implies choice penalty term appropriate words fmin far away tuning parameter chosen properly therefore aiming mimic fmin points right direction would like estimate following result provides upper lower bounds note permitted depend sample size assume remains upper bounded bounded away zero sample size increases however allow bounds unknown theorem assume conditions suitably chosen depending one therefore obtain probability least exp exp denote positive constants depending sample size theorem shows one obtain bounds choose tuning parameter properly since condition satisfied one obtains convergence probability ratio constant moreover theorem also provides bounds note hold probability tending one remark theorem one obtains min high probability sufficiently large sample size one say mimics noiseless counterpart fmin behaves noise observations therefore choice allows control random part problem using penalty term variance noise remark theorem one observe therefore able recover rate convergence estimation error illustrated section furthermore observe degree smoothness bounded probability although lower bound neither imply directly lower bound examples following require assumptions theorem hold example provide references results literature one verify condition satisfied refer interested reader details lower bounds one may first note additionally insert results yang barron authors show often global local entropies order example let smoothing spline estimator explicitly computed green silverman moreover shown case condition holds conditions design matrix kolmogorov tihomirov example van mgeer therefore theorem standard choice yields moreover obtain upper lower bounds high probability large remark recover optimal rate convergence stone consider case design points larger dimension let define index values integers write furthermore denote differential operator defined consider roughness penalization case condition holds birman solomjak obtain theorem choice similarly able recover optimal rate estimation error example example define total variation penalty denote design points let case condition fulfilled birman solomjak advantage total variation penalty example used unbounded let theorem choice also obtain bounds high probability large remark also recover optimal upper bound estimation error conclusions theorem derives concentration result around nonrandom quantity rather upper bound particular ratio convergences probability satisfies condition holds nonparametric setting suitably chosen shown theorem examples section strict concavity lemma plays important role derivation theorems work proof property requires square root penalty term convex furthermore proof theorems rely fact noise vector gaussian seen lemma invoke concentration result functions independent gaussian random variables lemma employ lower bound expected value supremum gaussian processes bound function proofs section divided two parts section first state prove lemmas necessary prove theorem follow closely proof theorem chatterjee however include roughness penalization term functions end section combine lemmas prove first theorem section first prove additional result necessary establish theorem present proof second theorem results literature used proofs deferred appendix proof theorem lemma strictly concave functions proof let take two values rmin define tvs take let trs properties seminorm implies moreover trs first inequality uses fact square root penalty term convex therefore using equations sup sup sup tvs tmn therefore concave taking expected value equations yields concave since strictly concave strictly concaves result lemma proof let sup show first suppose rmin note therefore contradiction definition must prove consequence definition inequalities minimize uniqueness follows lemma define moreover define event exp proof proof theorem chatterjee one easily observe applying inequality lemma therefore using lemma yields equations obtain ready prove theorem proof theorem let first note equal tends infinity rmin lemma know unique define event therefore construction moreover know definition since strictly concave lemma must combining lemma equation yields therefore lemma letting exp exp proof theorem proving theorem need following result lemma gives bounds unknown nonrandom quantity note bounds written parabolas maximums maximizers depending lemma assume condition let stated constants depending sample size rmin proof lemma proof makes use known results upper lower bounds expected maxima random processes found appendix indicate let rmin standard gaussian random variables define take two functions note follows gaussian distribution expected value variance therefore process respect metric index set see appendix note define diameter diamn supx difficult see diamn proceed obtain bounds dudley entropy bound see lemma appendix condition constants log sup moreover sudakov lower bound see lemma appendix log sup let diamn take last equation condition constant log log equations definition rmin writing completes proof ready prove theorem proof theorem condition know exist constants depending rmin take constants satisfying condition lemma equation first derive bounds let lemma recall rmin note rmin upper bound reached moreover know function attain negative values strict concavity lemma rmin must combining equation choice obtain rmin therefore following rationale substituting equation definition exist constant furthermore combining equations recalling rmin yields constant joining equations gives first result theorem proceed obtain bounds let constants exp first inequality used equation second theorem equation similarly exp first inequality use equation second theorem equation therefore exp exp second result theorem follows references birman solomjak piecewise polynomial approximations functions classes matematicheskii sbornik boucheron lugosi massart concentration inequalities nonasymptotic theory independence oxford university press chatterjee new perspective least squares convex constraint annals statistics del barrio deheuvels van geer lectures empirical processes theory statistical applications ems series lectures mathematics european mathematical society green silverman nonparametric regression generalized linear models roughness penalty approach chapman monographs statistics applied probability taylor francis smoothing spline anova models ima volumes mathematics applications springer kolmogorov tihomirov sets function spaces uspekhi matematicheskikh nauk koltchinskii oracle inequalities empirical risk minimization sparse recovery problems ecole springer stone optimal global rates convergence nonparametric regression annals statistics van geer empirical processes cambridge series statistical probabilistic mathematics cambridge university press van geer june estimating regression function annals statistics van geer wainwright concentration regularized empirical risk minimization preprint van geer wegkamp consistency least squares estimator nonparametric regression annals statistics van der vaart wellner weak convergence empirical processes springer series statistics springer wahba spline models observational data regional conference series applied mathematics society industrial applied mathematics yang barron determination minimax rates convergence annals statistics appendix lemma gaussian concentration inequality see boucheron let vector independent standard gaussian random variables let denote function following lemma applies gaussian concentration inequality show quantities close exponential probability exploited chatterjee lemma rmin proof let proof write sup let two standard gaussian random vectors properties supremum inequality sup sup sup sup therefore lipschitz second argument gaussian concentration inequality lemma every every rmin take applying gaussian concentration inequality yields combining last two equations yields result lemma next lemma need following definition stochastic process called respect semimetric index set every lemma dudley entropy bound see koltchinskii process respect following bounds hold numerical constant log sup sup log denotes diameter space lemma sudakov lower bound see boucheron let finite set let gaussian vector ext sup log moreover let defined smaller diameter lower bound rewritten sup log
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oct short survey bounding union probability using partial information jun yang fady alajaji glen takahara abstract short survey existing upper lower bounds probability union finite number events using partial information given terms individual pairwise event probabilities sums new proofs existing bounds provided new observations regarding existing bound given contents introduction review existing bounds observations bound references department statistical sciences university toronto canada department mathematics statistics queen university canada addresses jun takahara short survey introduction consider finite family events general probability space fixed positive integer note finitely many boolean specified probability interested bounding finite union events terms partial probabilistic event information knowing individual event probabilities pairwise event probabilities linear functions probabilities individual pairwise events example union upper bound bonferroni inequality respectively given follows note union upper bound established terms individual event probability actually needed however bonferroni lower bound estabp lished using two terms therefore union upper bound bonferroni inequality established based different partial information event probabilities order distinguish use different partial information assume vector represents partial probas bilistic information union specifically assume given integer denotes range function equals value function given example problem directly reduced finite probability space case thus consider finite probability spaces denotes elementary outcome instead atom short survey lower bound similarly upper bound established using partial information represented follows definition lower bound function set events value given equals note given multiple functions lower bounds example max max therefore need define optimal lower bound general class lower bounds functions let denote set lower bounds functions definition say lower bound optimal definition say lower bound achievable every inf infimum ranges collections represented bounds following lemma shows achievability equivalent optimality short survey lemma lower bound optimal achievable proof suppose achievable let given let lower bound achievability exist sets represented since holds prove converse contrapositive suppose achievable exists inf infimum ranges collections represented define satisfies inf larger hence optimal using lemma therefore prove lower bound optimal value one collection events represented optimal upper bound also defined similarly using supremum proved achievability example one easily verify following construction proof achievability optimal lower bound class maxi optimal lower bound class min optimal upper bound classes short survey furthermore prove lower bound optimal showing achievable example order show class bonferroni inequality optimal bound lower bounds functions need show achievable note lower bound negative values however according definition achievability lhs never negative means lower bound achievable therefore bonferroni inequality optimal throughout survey mainly focus lower bounds using different partial probabilistic information upper bounds presented remarks review existing bounds pwe start class lower bounds terms lower bound known optimal introduce lower bounds terms including caen bound kat bound next review lower bounds terms given including algorithmic stepwise lower bound bound finally existing upper bounds reviewed including hunter upper bound algorithmic greedy upper bound first define degree atom outcome finite probability space follows definition atom let degree denoted deg number contain therefore degree atom equals integer lower bounds using sidering note bonferroni inequality lower bound class however shown optimal means exists another function lower bound always sharper bonferroni inequality short survey defining deg one easily verify following identities note using equalities one derive lower bound simply via inequality equality holds particular outcomes union degree resulting lower bound written since short survey hence lower bound always sharper thanp shown optimal class reasonable since pthe lower bound established using inp formation however readily shown lower bound always sharper bonferroni inequality lower bound class bound known bound optimal lower bound terms denoting bound written denotes largest integer less equal first show bound solution linear programming problem following lemma lemma bound solution following problem min proof problem feasible solution exists objective function linear bounded optimal value objective function always attained boundary optimal attained least one vertices polyhedron formed constraints set feasible solutions lemma readily verified using fact one optimal feasible points problem vertex obtain vertex one need make inequalities active means two integers short survey satisfying min easily shown solution problem achieved thus solution bound existing proof optimality bound seen herein give alternative simpler proof proving achievable lemma bound optimal class lower bounds terms proof shown bound solution written recalling definition one construct two outcomes finite probability space consider following construction collection events otherwise always therefore bound achievable hence optimal note since bound optimal always sharper lower bound actually easily proved since lower bound lower bound objective function inequality using two constraints short survey lower bounds using section review lower bounds terms including kat bounds similar definition define deg one verify simplicity denote examine lower bounds functions one verify written linear functions follows kai caen bound similar lower bound using inequality kai summing one get bound follows noted caen lower bound always sharper bound short survey kat bound introduce kat bound solution problem given following lemma lemma kat bound solution following problem min kai proof one separate problem solve suboptimization problems separately min kai solved using method solving problem bound one see details alternative proof given solving dual problem shown kat bound always sharper bound bound furthermore dembo shown dem kat bound improves bound factor following lemma give alternative simpler proofs results lemma comparing bounds kat bound satisfies max short survey proof first substituting one get bound solution following problem min kai since every feasible point also feasible point problem relaxed problem therefore next easy show since based constraints one get bound lower bound objective function using inequality therefore lower bounded finally prove note bound given solution written aik aik suffices prove integer kai denoting one get first equality holds second equality holds therefore inequality active remark finally derive upper bound using maximizing problem kat bound short survey following problem max kai gives upper bound lower using lower terms individual event probaupper bounds bilities pairwise event probabilities seen special cases boolean probability bounding problem solved numerically via linear programming problem involving variables unfortunately number variables boolean probability bounding problems increases exponentially number events makes finding solution impractical therefore suboptimal numerical bounds proposed order reduce complexity problem example using dual basic feasible solutions hand bounds particularly important one apply existing bound using base bound optimize bound choosing optimal subset algorithmically note bound optimization via subset exploits full information examples bounds class includes stepwise algorithmic implementation kounias lower bound greedy algorithmic implementation hunter upper bound analytical bounds like kat bound also investigated works see short survey class bounds established arbitrarily chosen continuous set computed using resulting bound also exploits full information typical example bounds class gallotkounias bound see also kounias lower bound algorithmic implementation kounias lower bound bound written max subset set indices however computational complexity kounias lower bound exponential since exponential number subsets order reduce computational complexity algorithmic algorithm proposed using stepwise algorithm find index set maximizes rhs refer algorithmic implementation kounias lower bound stepwise lower bound bound let bound given kounias shown lemma vector range orthogonal null space result singular one choose subsets compute corresponding bound results bound rank corresponding therefore without loss generality wlog assume herein solution unique short survey bound written symmetric furthermore bound recently revisited authors shown bound reformulated max upper bounds upper bounds literature following introduce hunter bound algorithmic implementation greedy algorithm hunter upper bound algorithmic implementation hunter upper bound bound written max set trees spanning indices trees include indices nodes however computational complexity finding optimal spanning tree exponential via exhaustive search order reduce complexity one algorithmic algorithm proposed using kruskal greedy algorithm finding spanning tree weighted graph refer algorithmic implementation hunter upper bound greedy upper bound observations bound finally conclude survey two observations bound applying bound subsets events note many existing lower bounds fully explore available information improved algorithmically via optimization subsets however section prove bound improved applying subsets short survey lemma given bound solution following problem min aat proof always write satisfies particularly let cholesky sition orthogonal matrix solution aat first constraint implies aat therefore minimum achieved min theorem bound improved via optimization subsets proof denoting first constraint equivalent atn constraints ati second constraint equivalent constraints ati selecting subset resulting bound solution relaxed problem subset constraints since objective value relaxed problem must original problem bound using subset higher bound using full information iterative implementation bound theorem bound computed iteratively specifically denote bound using information short survey upper left submatrix btn btn matrix invertible last case never happens following inequality holds proof note matrix inverse lemma hermitian matrix btn btn substituting computed using since proved theorem inequality directly theorem recent works close survey referring reader recent works topic references behnamfar alajaji linder tight error bounds orthogonal block codes slow rayleigh flat fading ieee transactions communications references behnamfar alajaji linder efficient algorithmic lower bound error rate linear block codes ieee transactions communications boros scozzari tardella veneziani polynomially computable bounds probability union events mathematics operations research eprint http bertsimas tsitsiklis introduction linear optimization athena scientific chen seneta strengthening lower bound methodology computing applied probability caen lower bound probability union discrete mathematics dem dembo unpublished notes communicated sasson dawson sankoff inequality probabilities proceedings american mathematical society feng shen inequalities functional analysis combinatorics probability theory electronic journal combinatorics gallot bound maximum number random variables journal applied probability galambos simonelli inequalities applications springer series statistics probability applications springer isbn horn johnson eds matrix analysis new york usa cambridge university press isbn hoppe improving probability bounds optimization subsets discrete mathematics hoppe effect redundancy probability bounds discrete mathematics kuai alajaji takahara lower bound probability finite union events discrete mathematics references kuai alajaji takahara tight error bounds nonuniform signaling awgn channels ieee transactions information theory kounias bounds probability union applications annals mathematical statistics kuai tight bounds probability union applications signaling awgn channels thesis department mathematics statistics queen university mao cheng shen new lower bound error probability nonuniform signals awgn channels proc wireless communications networking conference ieee gao bounding probability union events aggregation disaggregation linear programs discrete applied mathematics new upper bounds probability events based graph structures rutcor research report yang alajaji takahara new bounds probability finite union events proc ieee international symposium information theory yang alajaji takahara bounding union probability proc ieee international symposium information theory ieee yang alajaji takahara bounding union probability using partial weighted information arxiv preprint yang alajaji takahara lower bounds probability finite union events siam journal discrete mathematics yang alajaji takahara bounding union probability using partial weighted information statistics probability letters
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sep whole genome phylogenetic tree reconstruction using colored bruijn graphs cole stanley anton paul quinn keith seth mark computer science department brigham young university provo utah usa department biology brigham young university provo utah usa computational biology institute george washington university washington usa email colelyman present kleuren novel method reconstruct phylogenetic trees using colored bruijn graph kleuren works constructing colored bruijn graph traversing finding bubble structures graph provide phylogenetic signal bubbles aligned concatenated form supermatrix phylogenetic tree inferred introduce algorithms kleuren uses accomplish task show performance reconstructing phylogenetic tree drosophila species kleuren reconstructed established phylogenetic tree accurately viable tool phylogenetic tree reconstruction using whole genome sequences software package available https algorithm whole genome sequence colored bruijn graph ntroduction whole genome sequences readily available affordable like never due advent highthroughput next generation sequencing ngs provided researchers vast amounts genomic sequencing data transformed landscape understanding genomes field phylogenetics discovers evolutionary relationship taxa exception transformation phylogenetics responded copious amounts high throughput data novel methods better suited handle large amounts data efficiently traditional phylogenetic methods traditional approach phylogenetic tree reconstruction requires homology search throughout genomes taxa multiple sequence alignment msa homologs tree construction resulting matrix steps computationally expensive may introduce many unnecessary assumptions avoided using method methods come without disadvantages one many methods abstract away source phylogenetic signal method akin shared propose whole genome phylogenetic tree reconstruction method using colored bruijn graph cdbg data structure commonly used detecting variation comparing genomes cdbg similar traditional bruijn graph dbg substrings certain length referred kmers sequence represent vertices dbg edge exists two vertices suffix first vertex prefix second vertex cdbg differs traditional dbg vertex associated unique color set colors could differing sample species taxon introduce kleuren dutch colors tribute nicolaas govert bruijn bruijn graph namesake software package implements methods kleuren works finding bubble regions cdbg one colors diverge node act regions taxa sequence taxon bubble extracted msa performed msa bubble concatenated form supermatrix phylogenetic tree evolution constructed ethods definitions given alphabet nucleotide codes let dbg defined set vertices ith unique sequence length set edges edge connecting two vertices sequence overlap characters let cdbg defined taxa dbg ith taxon refer distinct color taxon furthermore let path defined sequence vertices subsequences edge let bubble defined associated one colors first last vertices identical see figure finally let defined vertices unique kmers ith dbg software architecture use dbgfm software package construct represent dbg individual taxa kleuren provides interface interact individual dbg create cdbg taxon considered color dbgfm package uses space efficient representation dbg kleuren algorithms algorithm kleuren algorithm function kleuren bubbles bubbles initialized empty list colors endv ertex find ndv ertex color path extend path endv ertex color add path bubble end append bubble bubbles end end alignments bubble bubbles alignment multiple sequence alignment path bubble append alignment alignments end supermatrix concatenation alignments end function overall algorithm kleuren works iterating superset vertices discovering vertices could form bubble vertex could form bubble present colors set user command line parameter note lower bubble colored bruijn graph color color color path actgtg path actaggtg path actagtg paths bubble color figure example bubble colored bruijn graph colors taxa colors vertices represent following colors contain vertex color color contain vertex color contains vertex redcolor contains vertex color contains vertex example act startvertex gtg endvertex contained colors extended paths color startvertex endvertex potential bubbles may found kleuren take longer run vertices considered starting vertex bubble let considered starting vertex bubble end vertex found see section end vertex found path found color see section process repeated vertex either considered starting vertex bubble visited extending path starting ending vertex finding end vertex end vertex found traversing path startv ertex vertex found least colors endv ertex used function extend path see section algorithm find end vertex function function find ndv ertex startv ertex endv ertex endv ertex initialized empty string neighbors get eighbors startv ertex mpty neighbors mpty endv ertex neighbor neighbors colors endv ertex neighbor end end end return endv ertex end function extending path main functions discover sequences found bubble extend path functions see section extend path startv ertex endv ertex use recursive function traverses dbg color every possible path startv ertex endv ertex explored maxdepth provided command line parameter user maxdepth parameter allows user specify thorough kleuren search bubble higher maxdepth bubbles kleuren potentially find longer kleuren take depth exponentially potential paths traverse algorithm extend path functions function extend path startv ertex endv ertex color maxdepth path visited visited initialized empty set recursive path startv ertex endv ertex path color visited maxdepth return path end end function data acquisition measure effectiveness method used drosophila species obtained flybase chose group species thoroughly researched established phylogenetic tree function recursive path currentv ertex endv ertex path color visited depth maxdepth add currentv ertex visited depth maxdepth return alse end currentkmer endkmer return true end neighbors neighbors currentv ertex color neighbor neighbors neighbor visited continue end oldp ath path append suffix currentkmer path depth depth recursive path neighbor endv ertex path color visited depth maxdepth path oldp ath else return true end end end function tree construction parameters used dsk software package count kmers present drosophila species find bubbles used following parameters kmer size colors required contain vertex order search bubble starting vertex ran instances kleuren concurrently days find bubbles bubbles cdbg identified used mafft perform msa sequence every bubble kleuren identified see figure msa concatenated form supermatrix see figure using biopython phylogenetic tree inferred supermatrix maximum likelihood using iqtree see figure tree constructed used ete software package compare tree established one visualize trees bubble assumptions method based assumption bubbles representative homologous regions taxa genomes propose assumption reliable shown dbg suitable method align sequences identifying bubbles cdbg find sections graph contain phylogenetic signal iii esults kleuren constructed tree see figure consistent established tree found distance two trees even though color color color path act gtg path actaggtg path multiple sequence alignment sequences bubble figure color color color path act path path supermatrix multiple sequence alignments concatenated color color color phylogenetic tree figure multiple sequence alignment msa sequences bubble presented figure msa bubble concatenated supermatrix phylogenetic tree constructed resulting tree supermatrix inferred maximum likelihood ran many concurrent instances kleuren multiple days see section kmers explored potential bubbles meaning many bubbles could found cdbg would make phylogeny concrete final successful run number unsuccessful attempts made construct tree initial attempts unsuccessful kmers kleuren uses find bubbles segments file sorted kmers file lexicographic order vertices kleuren used search bubbles skewed towards vertices lexicographically first remedied issue shuffling order kmer file lexicographic bias towards bubbles kleuren finds previous attempt resulted tree normalized distance established tree occurred bubbles therefore enough phylogenetic signal correct tree constructed find bubbles split kmer file parts multiple instances kleuren could find bubbles concurrently also discovered high frequency adenines frequency around comparison nucleotides final supermatrix could skew final tree nucleotides differing mutation rates thought bias towards due fact recursivep ath function see algorithm neighbors may sorted function would traverse neighbor started traversing neighbors see algorithm line similar previous sorting problem shuffled order neighbors first neighbor traversed would always lexicographically first despite change final supermatrix produced true tree still bias towards see section onclusion introduced novel method constructing accurate phylogenetic trees using cdbg method kleuren uses whole genome sequences construct cdbg representation traverses cdbg discover bubble structures become basis phylogenetic signal taxa eventually produces phylogenetic tree ngs era progresses whole genome sequences becoming prevalent organisms phylogenies organisms never constructed kleuren viable method relatively quickly accurately construct phylogenies newly sequenced organisms uture ork plan optimize kleuren find bubbles shorter amount time replacing underlying data structure cdbg represented dbgfm current method used represent dbg kleuren sacrifices time efficiency memory efficiency storing entirely disk thus slowing queries dbg kleuren runs faster bubbles found phylogenetic signal present accurate tree constructed also plan investigate reasons high abundance supermatrix see section iii balance frequency nucleotides supermatrix furthermore would like look kleuren performs cdbg constructed using read sequencing data rather assembled genomes acknowledgment work funded utah nasa space grant consortium epscor byu graduate research fellowship authors would like thank kristi bresciano michael cormier justin miller brandon pickett nathan schulzke sage wright thoughts concerning project authors would also like thank fulton supercomputing laboratory brigham young university work maintain experiments run figure phylogenetic tree drosophila species constructed using kleuren tree resulted using kmer size required species contain vertex order algorithm search bubble starting vertex tree consistent established tree species eferences schuster sequencing transforms today biology nature methods vol dec online available https fan ives cannon assembly method phylogeny reconstruction sequencing data bmc genomics vol jul online available https luo hao cvtree phylogenetic tree reconstruction tool based whole genomes nucleic acids research vol web server jul online available https chan ragan phylogenomics biology direct vol jan online available https haubold phylogenetics population genetics briefings bioinformatics vol nov online available https steele bastola alignmentfree genetic sequence comparisons review recent approaches word analysis briefings bioinformatics vol jul online available https chan bernard poirion hogan ragan inferring phylogenies evolving sequences without multiple sequence alignment scientific reports vol sep online available https jin phylogenomic approach closely related organisms nucleic acids research vol jan online available https iqbal caccamo turner flicek mcvean novo assembly genotyping variants using colored bruijn graphs nature genetics vol jan online available https peng zhao novel bruijn graph algorithm gene construction unassembled transcriptomes genome biology vol nov online available https chikhi limasset jackman simpson medvedev representation bruijn graphs research computational molecular biology annual international conference recomb pittsburgh usa april proceedings sharan cham springer international publishing online available https ferragina manzini opportunistic data structures applications proceedings annual symposium foundations computer science ser focs washington usa ieee computer society online available http gramates marygold dos santos urbano antonazzo matthews rey tabone crosby emmert falls goodman ponting schroeder strelets thurmond zhou flybase consortium flybase looking future nucleic acids research vol oct online available https hahn han han gene family evolution across drosophila genomes plos genetics vol online available https rizk lavenier chikhi dsk counting low memory usage bioinformatics vol jan online available https katoh standley mafft multiple sequence alignment software version improvements performance usability molecular biology evolution vol jan online available https cock antao chang chapman cox dalke friedberg hamelryck kauff wilczynski hoon biopython freely available python tools computational molecular biology bioinformatics bioinformatics vol mar online available https nguyen schmidt von haeseler minh fast effective stochastic algorithm estimating phylogenies molecular biology evolution vol nov online available https serra bork ete reconstruction analysis visualization phylogenomic data molecular biology evolution vol feb online available https robinson dylus dessimoz interactive viewing comparison large phylogenetic trees web molecular biology evolution vol apr online available https raphael novel method multiple alignment sequences repeated shuffled elements genome research vol nov online available https minkin patel kolmogorov vyahhi pham sibelia scalable comprehensive synteny block generation tool closely related microbial genomes algorithms bioinformatics international workshop wabi sophia antipolis france september proceedings berlin heidelberg springer berlin heidelberg online available http minkin pham medvedev twopaco efficient algorithm build compacted bruijn graph many complete genomes bioinformatics sep online available https robinson foulds comparison phylogenetic trees mathematical biosciences vol feb online available https
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sending information status updates abdulrahman omur jing sennur aylin jan department electrical computer engineering university maryland college park department electrical computer engineering carnegie mellon university pittsburgh department electrical engineering pennsylvania state university university park consider energy harvesting transmitter sending status updates regarding physical phenomenon observes receiver different existing literature consider scenario status updates carry information independent message transmitter encodes message timings status updates receiver needs extract encoded information well update status observed phenomenon timings status updates therefore determine age information aoi message rate rate study tradeoff achievable message rate achievable average aoi propose several achievable schemes compare performances ntroduction consider energy harvesting transmitter sending status updates receiver via status update packets status update packet requires unit energy transmitter harvests energy stochastically time one unit time random order minimize age information aoi transmitter needs send frequent regular time status updates however frequency regularity updates constrained stochastic energy arrival process known causally transmitter paper different existing literature consider scenario timings status updates also carry independent message see fig order obtain tractable formulation consider abstraction physical channel noiseless transmitter battery unit size intuitively clarified shortly tradeoff aoi rate message goal paper characterize tradeoff scenario causal online knowledge energy arrivals determined order minimize average aoi transmitter needs apply threshold based policy exists fixed deterministic threshold energy arrives sooner seconds since last update transmitter waits sends update packet hand seconds since last update transmitter sends update packet right away energy arrives hand scenario considered capacity energy harvesting channel main challenge arises due channel state work supported nsf grants ccf cns energy requirements energy harvests normalized measurements message fig energy harvesting transmitter battery sends status updates independent information receiver energy availability introduced state due existence battery transmitter energy saved used later unavailability state information receiver reference converts problem regular channel uses timing channel obtains capacity terms auxiliary random variables using bits queues approach sending information necessarily requires transmitter send packet random amount time following energy arrival whereas minimizing aoi requires transmitter apply deterministic threshold based policy note transmitter sends packet either deterministic time energy arrival right time energy arrival thus send rate packet timings even though minimizes aoi main source tension aoi minimization information rate maximization subject paper paper first present general tradeoff region achievable aoi achievable information rate consider class renewal policies system action depends recent transmission within class policies first propose policies determine next transmission instant function time difference recent energy arrival recent status update consider simpler policies call separable policies policies separate update decision information transmission additive manner energy arrives transmitter decides update neglecting information transmission transmitter decides send update encodes message top update timing policies derive average achievable aoi achievable rate compare tradeoff regions policies observe numerically first class policies achieve better tradeoff regions also observe value energy arrival info symbol fig example evolution instantaneous aoi fig average energy arrival increases policies perform similarly related work minimizing aoi studied many different settings including settings energy constraints settings energy constraints offline online energy harvesting models energy harvesting communication systems extensively studied settings example offline scheduling settings considered online scheduling considered limits considered ystem odel consider noiseless binary energy harvesting channel transmitter sends status updates independent message simultaneously fig transmitter unit size battery energy arrivals known causally transmitter distributed according bernoulli distribution parameter hence times energy arrivals denoted geometric parameter transmission costs unit energy thus transmitter sends update battery depleted timings transmitted updates determine average aoi message rate instantaneous aoi given sending information timing channel minimum aoi given inf inf lim sup set feasible policies since transmitter equipped battery due energy causality note due memoryless property geometric distribution assume without loss generality time instant previous update time instant previous energy arrival send information timings status updates consider model studied section thus assume knowledge energy arrival instants causally transmitter receiver information time duration carried random variable see fig achievable rate timing channel lim inf lim inf sup sup second equality follows since denote tradeoff region tuple aoi achievable rate aoi minimum achievable aoi given message rate least achievable aoi inf lim sup time stamp latest received status update packet current time example evolution aoi shown fig average aoi defined lim sup lim inf sup lim sup denotes similarly duration two updates alternate characterization tradeoff region also total accumulated age two updates represented done using tuple achievable aoi area see fig expectation energy equal maximum achievable information arrivals possible randomness transmission decisions rate given aoi iii achievable radeoff egions update channel use immediately energy arrival section consider several achievable schemes considered achievable schemes belong class renewal policies renewal policy policy action time function current energy arrival instant average aoi renewal policies lim sup energy timing adaptive transmission policy etatp policy information carried random function energy arrival realization general case renewal policies optimal tradeoff obtained solving following problem results renewal reward theory theorem since use renewal policies hereafter drop subscript random variables maximum achievable information rate reduces max min max next present achievable schemes first scheme information transmission adapted timing energy arrivals takes long time energy arrive transmitter tends transmit less information energy arrives early transmitter tends transmit information scheme fully adapts timings energy arrivals comes cost high computational complexity relax adaptation two regions divided threshold energy arrives less slots transmit information using geometric distribution parameter energy arrives slots transmit information using another geometric random variable parameter choice geometric random variable hereafter motivated fact maximizes information rate energy arrival timings known receiver see section previous schemes instantaneous information rate depends timings energy arrivals next relax assumption assume instantaneous information rate fixed independent timings energy arrivals call policies separable policies policies transmitter two separate decision blocks first block status update takes decision depending timing energy arrival second block encoding desired message top timings updates similar spirit coding first separable policy update decision threshold based function inspired energy arrives threshold update block decides update energy arrives update block decides update immediately information block generate update immediately adds geometric random variable carry information timing top timing decided update block second separable policy call policy update block decides maximum possible value equal maxp solution problem found considering following alternative problem gives tradeoff region aoi reduces fixed problem concave solved efficiently obtain entire tradeoff region sweep possible values parameter possible values aoi solution found numerically optimizing possible conditional pmfs value use line search search optimal repeated possible values aoi finding optimal solution high complexity hence propose following policy reduces complexity significantly time adapts timing energy arrivals extent possible within set policies simplified etatp policy simplify form dependence transmission timings energy arrivals significantly transmitter waits energy arrives energy takes slots since last update transmit information using geometric random variable probability success otherwise transmitter transmits information using geometric random variable probability success transmitter chooses follows case variables optimization performed average achieved information rate function obtained equal calculate average aoi policy finally note case equal schemes simpler general class etatp still need search optimal reduce complexity next policy threshold based transmission policy present first separable policy policy assume information still carried see fig duration transmitter decides wait order minimize aoi duration transmitter decides wait add information timing update independent implies duration determined according threshold policy follows optimal value yet determined optimization variable optimal value calculated thus known transmitter receiver hence threshold policy deterministic policy ensures still consistent choose geometric random variable parameter tradeoff region written min fixed positive number feasible values equal follows smallest value take equal optimization problem case becomes function need calculate calculate follows calculate follows substituting quantities optimization problem solving jointly gives solution transmission policy policy similar threshold based policy one difference update block wait energy arrives instead decides update right away hence tradeoff region obtained solving min calculate independent message independent energy arrivals since geometric optimization problem function single variable problem solved line search umerical esults compare tradeoff regions resulting proposed schemes plot regions figs different values average energy arrivals namely low values fig significant gap performance etatp simplified schemes value region simplified etatp performs better threshold policies value increases shown fig fig gap performance different policies decreases significantly fig threshold policies overlap fig simplified etatp threshold policies overlap cases policy performs worst consistent early results early results context energy harvesting recent results updating soon one optimum eferences yang optimal status update age information minimization energy harvesting source ieee trans green communications networking appear also available online tutuncuoglu ozel yener ulukus binary energy harvesting channel battery ieee trans info theory april anantharam verdu bits queues ieee trans info theory january kaul yates gruteser status often one update ieee infocom march fig tradeoff region fig fig tradeoff region kaul yates gruteser status updates queues ciss march yates kaul status updating multiple sources ieee isit july kam kompella ephremides age information random updates ieee isit july costa codreanu ephremides age information packet management ieee isit june sun yates koksal shroff update wait keep data fresh ieee infocom april kosta pappas ephremides angelakis age value information age case available bedewy sun shroff information updates multihop networks available parag taghavi chamberland status updates symbol erasure channels ieee wcnc pages march yates najm soljanin zhong timely updates erasure channel ieee isit june yates lazy timely status updates energy harvesting source ieee isit june bacinoglu ceran age information energy replenishment constraints ucsd ita february arafa ulukus age minimization energy harvesting communications delays asilomar october arafa ulukus transmission energy harvesting networks ieee globecom december yang ulukus optimal packet scheduling energy harvesting communication system ieee trans january tradeoff region tutuncuoglu yener optimum transmission policies battery limited energy harvesting nodes ieee trans wireless march ozel tutuncuoglu yang ulukus yener transmission energy harvesting nodes fading wireless channels optimal policies ieee jsac september zhang optimal energy allocation wireless communications energy harvesting constraints ieee trans signal september yang ozel ulukus broadcasting energy harvesting rechargeable transmitter ieee trans wireless february yang ulukus optimal packet scheduling multiple access channel energy harvesting transmitters journal comm networks april wang aggarwal wang iterative dynamic fading channels energy harvesting ieee jsac march tutuncuoglu yener optimal power policies energy harvesting transmitters interference channel journal comm networks april gunduz devillers general framework optimization energy harvesting communication systems battery imperfections journal comm networks april wang liu simplicity meets optimality efficient transmission power control stochastic energy harvesting ieee infocom april khuzani mitran online energy harvesting multiple access communication systems ieee trans info theory february shaviv ozgur universally online power control energy harvesting nodes ieee jsac december baknina ulukus optimal online strategies energy harvesting broadcast channels ieee jsac december inan ozgur online power control energy harvesting multiple access channel wiopt may baknina ulukus online policies multiple access channel common energy harvesting source ieee isit july ozel ulukus achieving awgn capacity stochastic energy harvesting ieee trans info theory october mao capacity analysis discrete energy harvesting channels ieee trans info theory september shaviv nguyen ozgur capacity channel finite battery ieee trans info theory november jog anantharam geometric analysis awgn channel constraint ieee trans info theory august ross stochastic processes volume john wiley sons new york
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rohlin distance evolution influenza virus weak attractors precursors dec raffaella riccardo mario dipartimento fisica infn parma parma italy department biochemistry university zurich zurich switzerland dipartimento fisica infn parma parma italy corresponding abstract evolution hemagglutinin amino acids sequences influenza virus studied method based informational metrics originally introduced rohlin partitions abstract probability spaces metrics require previous functional syntactic knowledge sequences sensitive correlated variations characters disposition efficiency improved algorithmic tools designed enhance detection novelty reduce noise useless mutations focus usa data usa data show clusterization distance matrix gives strong evidence structure domains sequence space acting weak attractors evolution good agreement epidemiological history virus structure proves robust respect variations clusterization parameters extremely coherent restricting observation window results suggest efficient strategy vaccine forecast based presence precursors buds populating recent attractor introduction long history approaching dna rna sequences texts quantitative estimates various kinds statistical properties complexity indicators general idea behind approach information encoded sequence strictly related properties corresponding biological structures good indicators able recognize similar functions different sequences sector devoted estimate relevance mutations along time ordered set evolving sequences particularly interesting sufficiently long record samples accessible viral rna rapidly evolving diseases focus definite kind statistical properties precisely metric properties distance sequences concept admitting several implementations context evolving viral rna distances based sequence symbols mostly hamming type two strings characters hamming distance number sites different symbols distances sensitive local features since mutations occurring different sites non correlated framework interesting solution proposed focusing particular locations sequence epitopes whose role peculiarities influenza virus evolution well known authors succeeded extracting important features strains evolution sense extra information introduced choice epitopes proved efficient overcoming intrinsic uncorrelation hamming metrics leading interesting results relevant approaches based sequences information rely entropic distances referred shannon entropy compression algorithms mainly addressed comparison strings different length inhomogeneous frameworks procedure motivated fact evolving sequences beside substitutions frequent insertions deletions remarkable alternatives sequence based type distances hemagglutination inhibition assays reporting ability ferret antibodies raised one viral strain inhibit second strain ability agglutinate red blood cells currently used define similarity antigens certainly metrics extracted tests directly related real antigenic similarity strains requires assay animal data difficult obtain high precision intend introduce enhanced version different metrics known rohlin distance based sequences symbols expected sensitive global distributions correlations also founded shannon entropy differently informational functionals applies biologically homogeneous framework moreover deal frequency probabilities symbols single sites poor units touch global structure see however interesting improvements directions approach basic entities indeed partitions sequence subsequences determined starting configuration list amino acids projection operation precisely consider partitions defined homogeneous segments aims evidencing ordered collection connected subsequences equal symbols instance alphabetical string aaabbacccb would divided five subsequences subsequence would determine segment first segment aaa correspond site subset labels subset etc natural length segment number symbols example lengths lengths correctly normalized assignment equivalent definition probability measure subset algebra proportional number sites contained subset finally partitions represented bounds segment extremes economically left extremes allowing simple straightforward comparison partitions partition space rohlin distance defined couple partitions mutual conditional shannon entropy see materials methods details conditional shannon entropy represents residual information needed describe segment disposition disposition known terms knowledge may contribute knowledge therefore symmetrized form defining total information required distinguish seen schemes segments probabilities absence bias choice assigning equal weight site leading measure proportional length natural probability measures defined depend configurations since case set would different measures two configurations conditional shannon entropy would loose meaning inhibiting definition rohlin distance segmentation provided partitioning sequences homogeneous subsequences entails many advantages definition simple universal sequences tightly fractioned single symbols priori knowledge required along exigence black box analysis moreover even segments intrinsic biological meaning could appear inconvenience alterations overall distributions emerging historical records definition compatible biologically efficient features proposed evolution thus using probabilities arise geometrical topological asset distance measures information content carried evolution giving evidence emerging dissimilarities clearly content filtered far possible effects non evolving part end introduce called reduction process method designed amplify relevant differences partitions dropping common means amplified distance practical point couple reduced couple view reduction consists erasing common extremes segments two partitions key point method proves surprisingly effective also filtering noise useless mutations details given material methods section shall deal interesting example highly mutating sequence rna influenza virus whose databases particularly rich precisely consider amino acids sequence surface protein hemagglutinin subtype influenza virus human host strains collected usa analogous sequence subtype strains collected usa method works equal length strings based informational metrics long range correlations expected perform better applied longest sequences therefore choose full length sequences subsample available sequences also consider sequences identified complete date time sequence appears represents important information analysis restriction usa sequences motivated several facts first choosing sequences temperate regions give relevance seasonal timing virus evolution minimizing interference dephased development second geographical bounds sampling ensure looking season season reasonably stable population notwithstanding restrictions example process sequences northern hemisphere world words keep statistical majority disregarding possible noise analogous estimates hold processed sequences partitioning sequence calculate rohlin distance partition pairs analyze whole sequences sample hierarchical complete linkage clustering algorithm distance matrix procedure strong analogies analysis presented completely different metric space namely partition space instead configuration space analysis traces evolution topological properties sequences virus escapes following antigenic drift interestingly structures arising sequence space according metrics result quite meaningful individuate indeed well defined regions sequences space acting weak attractors evolution virus takes place definite periods moreover attractors display precursors sequences populating regions well identified circulating strains information appears relevant forecast problem suggesting alternative strategy vaccines formulation analogies also differences privileged regions evolution presented previous analysis attractors arise indeed similitude related overall disposition homogeneous segments actual data lying every site supplementary restriction epitopes means overlap results two approaches present complementary points view enforcing one another therefore quite remarkable vaccine predictions two methods agree well fact whose origin completely understood yet results clustering weak attractors rohlin metrics using sequences set calculate whole matrices hamming rohlin distances whose entries respectively sequences hik rik reduced rohlin distance corresponding partitions possibly simplified reduction process primitive partitions result far less informative reduced ones shall omit report calculations fact non trivial result principle deleting common bounds reduction process could imply indeed survival random set unstable bounds created incoming mutations happens expected artificially created random mutations reported last subsection clearly far informative chaotic residual set would useless scope empirical fact observe quite different reduction process cleaned useless common surviving bounds defining new reduced partitions far random carry winning novelty adaptive strategy virus terms effective new disposition segments random selected moreover mutual disposition intrinsically long range character captured non local metrics rohlin distance details amplification given mathematical section materials methods used clustering tool scenario emerges clustering analysis matrix standard hierarchical complete linkage algorithm number clusters external parameter important point therefore choice optimal since fixed every clusterization corresponds partition whole set groups populated sequences defining probabilities clusterization associate shannon entropy log probability distribution since growing cluster populations increase entropy however substantially stable means clusters also stable equivalently newly added probabilities small grows clusters probabilities almost continuously splitted smaller smaller ones observing behaviors rohlin hamming entropies figure note clear rohlin interval hamming always growing latter interpretation hamming clusters artificial product procedure split remarkable continuity rohlin long plateau clearly indicates clusters real structures sequences space keeping definite individuality large observation range growth low simply due fact imposed number small calculated clusters must contain real ones growing splitting effective optimal number calculated real clusters coincide contrary large calculated clusters numerous also real ones begin split much effectively peripheral loss registered plateau plateau extremes may therefore roughly related typical isolation length among real clusters maximal diameter respectively interestingly optimal value obtained clustering without additional hypotheses consistent number different circulating strains identified tests analogous results obtained shown figure case optimal value clustering parameter since every sequence marked sampling date natural question time distribution resulting clusters upshots summarized figure part refers comparison clustering hik scale axis clustering ordinate distinguishes among clusters ordinate cluster order color intrinsic meaning chosen readability criteria different polygonal symbols represent reference viruses including alternates observed northern hemisphere according tests names indicated plot details sequences given table reference sequences identified time coordinate processed together dataset positioned clusters belong rohlin clustering procedure vertical lines separate winter seasons conventionally set july clustering figure stable changing plateau range figure draw several indications first clear long temporal extension clusters densely populated several winter seasons interestingly present precursors term buds successors bunch sequences representing viruses appear time main part cluster identification buds explained details next sections let consider instance season year observe bud wuhan strain yellow cluster getting stronger following season living jointly johannesburg strain red cluster becomes dominant strain may considered successor sidney light green dominant strain notice sidney bud distribution clusters suggests evolution sequence space takes place preferential regions corresponding cluster populated well main season regions act kind weak non definitive attractors example mentioned two virus strains circulated winter season wuhan successors sydney buds tests revealed wuhan reference virus circulating one recommending vaccine season crucial already analysis shows emergence bud sydney family strain actual virus circulated winter season rohlin attractors correctly describe also heterogeneity coming outliers sequenced must treated separately happened hamming based approaches sequences naturally fall cluster confirming rohlin correctly takes account variability present outliers second point consistent epidemiological data example subsequent strains appeared years according tests represented three well defined clusters moscow alternate interestingly reference sequences belong correct clusters included data set without priori information lower part figure clustering procedure shown referring hamming matrix hik definite temporal extension clusters observed agreement previous results however cluster temporal distribution obtained hik quite unstable confirming dependence evidenced figure used rohlin choice completely arbitrary clear plateau hamming means appearing spots true buds results almost continuous splitting hamming clusters stable change namely new cluster produced simply raising moreover seasons contrast indication cluster arrangement clusters represented reference sequence others wrongly doubly represented showing poorer correlation analysis example sydney moscow reference strains belong cluster expected different ones buds rohlin weak attractors vaccine forecast interesting evidence drawn position symbols attractors symbols upper diagram figure show first row vaccine indicated basis tests previous seasons second row represent indication would suggest basis following criterion looking distribution year simultaneous strains statistically significant populations would indicate newest one bud emergent next year criterion words sees novelty carried emergent buds feature enhancing aggressiveness virus buds analysis symbols shown green vaccine choice agrees circulating strain red used symbol one virus circulated season corresponding prediction agrees one two circulating strain second symbols row indicates buds criterion able identify correctly circulating strain every year apart season lack sequences criterion fails cases course extremely interesting verify criterion set sequences next season soon available lower part figure displaying results procedure hamming distance note buds early warning new strains reliable discussed instability allow unambiguous detection appearance produced raising value bud criterion successfully applied also interesting case results shown figure sampling inhomogeneous statistics increases limit analysis period clustering entropy analysis shown figure optimal relevant cluster starting suddenly around april red line figure represents precisely pandemic virus appeared season case bud criterion partially fails recognizes correctly strains circulated beginning season reasonable since method expected effective simple antigenic drift case dramatic change pandemic case shift probably sets completely new direction sequence space notice method evidences simultaneous occurrence four well distinct clusters feature missing hamming analysis observation multiple clusters signals could related typical instability periods one facing according expect structure present also new set sequences season restricting time window another natural question relevance examined time window treated figure displaying results one would obtain stopping data collection five different years applying clustering restricted sets sequences available times procedure intended clarify bud criterion works check unpredictive posteriori verification vaccine choice real working framework time distribution axis describes various years position sequences exactly upper figure however reproduce exactly situation vaccine prediction made collect sequences available march given year perform clustering analysis dataset therefore increasing number sampled sequences every horizontal time sector starting upper part lowest details apply entropy criterion time dataset restricted end winter season given year dataset choose optimal case interestingly clear plateau entropy analysis every restricted time window allowing unambiguous choice number clusters rohlin entropy analysis restricted time window shown figure together corresponding one hamming notice hamming clustering clear indication number clusters entropy analysis evidence emergent buds principle leaving part data clusterings could different final one remarkable structure remains buds clearly present evolution took place well defined landscape preferential antigenic directions filled genetic drift acting therefore weak attractors symbols reference strains excluded clustering time restricted window would represent posteriori knowledge associated cluster inverse analysis calculating reference strain minimum distance sequences belonging bud cluster details reverse analysis applied also whole dataset given next subsection figure syringe indicates accredited vaccine next year based bud criterion even previous years database poor forecast good cluster corresponding syringe exactly prevailing cluster observed next year confirming coherence procedure result supports bud emergence criterion prediction new prevailing strain interestingly bud criterion include additional information epitopes positions agrees well dominant strain prediction discussed appears indicates black box rohlin distance analysis able grasp biological information included epitopes metrics certainly correct requires additional input relevant positions notice results vaccine choice always agree prediction single circulating strain complementary two clear explanation interesting fact moment testing method reverse analysis random permutations clustering procedure performed completely different method consider distances sequences hierarchical method refers different sequences identified analysis precisely rohlin distance calculated sequences reference virus strains sequences temporally aligned along axis figure reference strains conventional position along axis diagram sequence represented point whose coordinate sampling date whose coordinate corresponds nearest reference strain surprisingly final result new procedure almost one showed figure main text obtained completely different analysis analogous plot hamming distance preserve cluster structure rohlin distance approach proves robust consistent tests analysis check robustness results consider random permutation site labels simultaneously performed sequences dataset operation leaves hamming distances invariant definition since random mixing amino acids biologically meaningless natural request conclusions drawn correct metrics crash precisely happens rohlin distance words rohlin partitions corresponding real sequences seem encode correctly antigenic drift evolution evidencing meaningful relation global structure sequences vice versa simple global mutations counting completely fails recognize information deletion caused label permutation results presented figure discussion mechanism underlying influenza antigenic plasticity virus continually escapes immune system producing variant strains cause within years remains outstanding evolutionary problem two main general pictures evolution first one based almost continuous slow drift ancestor sequence large shifts occurring certain stages evolution second one relies punctuated evolution antigenic swarms sequences populate basins sequence space several years circulating swarm jumps another basin reinfecting population dynamical model type evolution built evidence punctuated antigenic evolution put forward several authors picture emerging rohlin metrics seems support punctuated evolution better fit epidemiological data giving also insights relevant distance circulating strains vaccines fact clusters result organized well defined regions sequences space virus appears explore several seasons sequence space region corresponding certain rohlin width jump takes another attractor evolution starts return previous region also possible regions constitute therefore weak attractors sense able trap virus finite time also years words weak attractors seem identify privileged antigenic directions genetic data interestingly clusters present precursors termed buds small number sequences explore advance next attractor strains still belong previous one goes precursors manage experience winning escape strategy followed main swarm subsequent years clear correlation emerges bud younger attractor appeared given year circulating strain subsequent season bud criterion parallel analysis could helpful correct choice vaccines picture emerging rohlin distance analysis appears hold also processing analogous data sets usa interestingly bud criterion partially fails recognizes correctly emerging bud pandemic period able predict clear new cluster appears suddenly april analysis correctly signals also high instability phase conclusion main points stressed first priori knowledge biological nature used put data set indications derived clustering distance matrix constitute genuine emergence seems plausible therefore approach could work similar circumstances homogeneous set equal length arrays disposal second point existence structures sequence space described weak attractors evolution viral species takes place discontinuous dynamics clearly clustering algorithm expected recognize chronological order within distance matrix whenever distance monotone function time case one would also expect progressive fragmentation clusters external parameter grows substantially behavior suggested analysis hamming matrix contrary presence precursors discontinuously anticipate onset future attractors stability attractors structure varying sampling quite non trivial facts implying rohlin attractors conventional decomposition sequence space possess instead robust intrinsic natural meaning seems therefore rohlin distance reduced couples able evidence selected variety admissible antigenic states preferentially explored mutations remains hidden metric approaches third point buds emergence criterion could offer valuable complementary tool optimal strategy choice vaccines matter obviously delicate long series experimental checks explore confirm possibility practical utilization think effort direction worthwhile materials methods data main database reference constituted proteins subtype influenza virus isolated usa excluding sequences incomplete sampling date set enriched reference sequences corresponding reference viruses circulated years according analysis total sequences written amino acids alphabet main database reference constituted proteins subtype influenza virus isolated usa excluding sequences incomplete date set enriched reference sequences corresponding reference viruses circulated years according analysis total sequences written amino acids alphabet sequences rohlin metrics let finite alphabet characters amino acids case sequence may thought function one dimensional array sites labeled function defines configuration state every sequence therefore element configuration space probability measure finite subset algebra given normalized number sites every subset means sites assumed equivalent instance subset includes sites measures possible assigning weights sites however weights measures depend configurations otherwise subset could one measure simultaneously functionals defined would loose meaning partition exhaustive collection disjoint subsets called atoms space set possible partitions partial order means refines product close analogous minimal common multiple minimal partition refining factors unit partition one atom whole set obvious properties easily follow every operation maximal common factor refined common clearly etc every partition may thought experiment elementary atomic event occurs probability meaning definitions trivial experiment means etc shannon entropy defined another partition conditional entropy quantities give respectively mean incertitude experiment residual mean incertitude result known rohlin distance useful formula follows thus measure overall giving account mutual correlations among respective outcomes concepts definitions hold true probability spaces discrete spaces graphs lattices states configurations determined values assumed sites finite alphabet therefore deeply different well known hamming distance configurations distance defined possible normalization factor distance alphabet numerical otherwise case since alphabet amino acids counting sites different symbols regardless position tells one nothing correlations mutations important stress hamming rohlin distances defined objects former configurations latter partitions particular case finite lattice states configurations character sequences length shall work partitions generated homogeneous segments consecutive sites value course exist much partitions non connected atoms example consider fictional configuration atoms indicated site labels corresponding partition map univocal non invertible since several configurations mapped partition instance mutation affect boundaries leaves segment structure unchanged thus correspondence rohlin distance evaluate different states regard correlated distribution segments true loss information due projection many configurations partition comparable loss takes place also hamming since single site contribute gives account distinction moreover noticed sites always totally uncorrelated two partitions could confused weakened presence tight common factor would eliminate far possible order amplify rohlin distance giving evidence real emerging novelty however reduction operation analogous reduction minimal terms fractions uniquely defined partitions admit unique factorization primes role prime indecomposable factors played dichotomic still extremely redundant key point consists defining partition restricted family elementary dichotomic factors following features must well defined every least subset actually investigation contain factors number atoms assuming elementary factors families defined reduction process consists following steps define maximal common divisor drop factors relatively prime note surviving factors respectively define words drop dichotomic factors subfactors maximal common factor generated surviving factors amplification reduced consequence following property proposition indeed proof elementary recalling write mentioned contains factors dropped reduction therefore formula rephrased fact thesis moving terms sides using formula conditional entropy thesis reduces clearly true since conditioning terms greater left sides important remark amplification regards couple whole reduced partitions single entities complexity possibly decreases expected means etc defining reduction process idempothe correspondence tent projection subset irreducible pairs process therefore essentially depends family elementary factors choice priori implemented many ways reflecting kind interest observer experiment details procedures abstract probability spaces may found sketch algorithmically easy recipe fitting special case character strings exploiting partition segments connected subsequences economically represented list left bounds segments example fully determined suggest convenient choice family elementary factors precisely every dichotomic factor therefore terms labels example gives etc choice reduction process described consists erasing common labels apart first one label indeed necessarily common bound alignment instance consider new configuration list list list reduced represented respectively note correspond new sequences since reduction performed directly graphic intuitive representation reduction given figure acknowledgments work partially supported miur firb grant infn project biological applications theoretical physics methods authors thank roberto burioni nicola clementi useful discussion suggestions references waterman introduction computational biology chapman hall crc press pevzner computational molecular biology algorithmic approach mit press cambridge hampson influenza potter elsevier london hoftand belshe genetic archaeology influenza new england journal medicine nelson holmes evolution epidemic influenza nat rev genet liao lee hsiung bioinformatics models predicting antigenic variants influenza virus bioinformatics hamming error detecting error correcting codes bell system technical journal gupta earl deem quantifying influenza vaccine efficacy antigenic distance vaccine deem clustering detects incipient dominant influenza strain clusters prot eng des sel otu sayood new sequence distance measure phylogenetic tree reconstruction bioinformatics xia jin zhu zhou using mutual site transition network map genetic evolution influenza virus bioinformatics ouyang zhu wang multivariate entropy distance method prokaryotic gene identification bioinform comput biol butte kohane mutual information relevance networks functional genomic clustering using pairwise entropy measurements pac symposium biocomputing rao rodriguez benson evaluating distance functions clustering tandem repeats genome informatics baxevanis ouellette bioinformatics practical guide analysis genes proteins wiley new york hirst studies antigenic differences among strains influenza means red cell agglutination exp med smith lapedes jong bestebroer rimmelzwaan mapping antigenic genetic evolution influenza virus science arnold avez ergodiques classique paris martin england mathematical theory entropy reading sun deem spontaneous emergence modularity model evolving individuals physical review letters wolf nikolskaya cherry viboud koonin projection seasonal influenza severity sequence serological data plos curr bao bolotov dernovoy kiryutin zaslavsky influenza virus resource national center biotechnology information virol hastie tibshirani friedman elements statistical learning springer new york plotkin dushoff levin hemagglutinin sequence clusters antigenic evolution influenza virus proc natl acad sci usa khinchin mathematical foundations information theory dover new york morens taubenberger fauci pandemic influenza virus next mbio fitch bush bender cox long term trends evolution human influenza type proc natl acad sci usa koelle cobey grenfell pascual epochal evolution shapes phylodynamics interpandemic influenza humans science pybus rambaut evolutionary analysis dynamics viral infectious disease nat rev genet recker pybus nee gupta generation influenza outbreaks network host immune responses limited set antigenic types proc natl acad sci usa holmes ghedin miller taylor bao analysis human influenza virus reveals multiple persistent lineages reassortment among recent viruses plos biol data obtained niaid ird online web site http agliari casartelli vivo metric characterization cluster dynamics sierpinski gasket stat mech casartelli dall asta rastelli regina metric features dipolar model phys math gen figure legends figure looking optimal clustering clustering entropy hamming rohlin distance different values plateau rohlin suggests optimal stable result clustering figure clusters time evolution upper part rohlin clusters time evolution lower part hamming clusters time evolution reference sequences shown corresponding symbols names details indicated table upper part indicate vaccine choice according indication buds criterion lower green red colors indicate right wrong choice respect corresponding analysis real circulating strain double color used one strain circulated year corresponding prediction agrees one circulating strain figure clusters time evolution upper part rohlin clusters time evolution lower part hamming clusters time evolution reference sequences shown corresponding symbols indicated table upper part indicate vaccine choice according indication buds criterion lower green red colors indicate right wrong choice case symbols correspond specific test indicated star pentagon notice onset pandemic virus failure bud criterion line figure changing time window rohlin analysis years clusters structure robust sampling increase buds appeared seasons correctly reveal future circulating strains indicated syringe figure reduction two partitions derived sequences given text maximal common factor thick vertical lines individuate atomic segments reduced colors remind source configurations colors partitions partitions originate sequences symbols directly defined partition space tables table symbols legend fig table symbols legend fig supporting information legends figure looking optimal clustering clustering entropy rohlin hamming different values influenza long plateau rohlin suggests stable well defined value optimal notice hamming growing figure looking optimal clustering restricted time window clustering entropy rohlin hamming different values influenza obtained considering sequences end winter season year indicated plot time window long plateau rohlin suggests stable well defined value optimal figure correspondence fig main text figure reverse analysis rohlin clusters sequences minimum distance corresponding reference sequences years great similarity fig shows strong consistency rohlin analysis figure clustering random permutations effect random permutation symbols entropy clustering function indicates entropy clustering rohlin distance stands entropy clustering sample obtained random permutation symbols sequence
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finding largest common substructures molecules quadratic time andre droschinsky nils kriege petra mutzel oct dept computer science technische dortmund germany abstract finding common structural features two molecules fundamental task cheminformatics drugs small molecules naturally interpreted graphs hence task formalized maximum common subgraph problem albeit vast majority molecules yields outerplanar graphs problem remains consider variation problem high practical relevance rings molecules must broken block bridge structure input graphs must retained common subgraph present algorithm finding maximum common connected induced subgraph two given outerplanar graphs subject constraint approach runs time outerplanar graphs vertices maximum degree leads quadratic time complexity molecular graphs bounded degree experimental comparison synthetic datasets shows approach highly efficient practice outperforms comparable algorithms introduction maximum common subgraph problem arises many application domains necessary elucidate common structural features objects represented graphs cheminformatics problem extensively studied often referred maximum largest common substructure problem two variants problem distinguished maximum common induced subgraph problem mcis find isomorphic induced subgraphs two given graphs largest possible number vertices maximum common edge subgraph problem mces require common subgraphs induced aims maximizing number edges variants reduced maximum clique problem product graph two input graphs cheminformatics mces used frequently since reflects notion chemical similarity adequately reduce running time product graph based algorithms although algorithms still exponential running time worst case commonly applied molecular graphs practice work supported german research foundation dfg priority programme algorithms big data spp droschinsky kriege mutzel however several restricted graph classes render polynomial time algorithms possible seminal work direction attributed edmonds proposed polynomial time algorithm maximum common subtree problem given graphs desired common subgraph must trees recently shown problem solved time unrooted trees vertices maximum degree induced subgraph isomorphism problem decide pattern graph isomorphic induced subgraph another graph generalized mcis mces respectively variants even pattern forest graph tree pattern tree outerplanar hand graphs biconnected outerplanar induced solved time complexity results demand cheminformatics lead consideration mces block bridge preserving bbp constraint requires common subgraph retain local connectivity input graphs computable also yields meaningful results cheminformatics algorithm recently proposed requires time outerplanar graphs mentioned polynomial time algorithms either applicable molecular graphs impractical due high constants positive exception approach shown outperform algorithms molecular graphs practice algorithm stated running time fact leads running time worst case contribution take concept bbp propose novel bbpmcis algorithm running time outerplanar graphs vertices maximum degree obtain result combining ideas maximum common subtree problem new algorithm biconnected mcis biconnected outerplanar graphs subproblem develop quadratic time algorithm exploits fact outerplanar embedding biconnected outerplanar graph unique moreover algorithm allows list solutions quadratic total time approach supports solve weight function mapped vertices edges experiments show almost cases yields results molecular graphs adequate weight function method outperforms terms efficiency approach orders magnitude preliminaries consider simple undirected graphs let graph refer set vertices set edges edge connecting two vertices denoted order graph number vertices let graph finding largest common substructures molecules quadratic time called induced subgraph write graph connected path two vertices connected component graph maximal connected subgraph graph called biconnected connected maximal biconnected subgraph graph called block edge contained block subgraph called bridge vertex called cutvertex consists connected components graph planar admits drawing plane two edges cross connected regions drawing enclosed edges called faces unbounded region referred outer face edge face said incident edge touches face two faces adjacent incident common edge graph called outerplanar admits drawing plane without crossings every vertex lies boundary outer face matching graph set edges two edges share vertex matching maximal matching perfect weighted graph graph endowed function matching weighted graph weight maximum weight matching mwm matching isomorphism two graphs bijection common induced subgraph isomorphism isomorphism induced subgraphs subgraph block bridge preserving bbp bridge bridge two edges different blocks different blocks common subgraph isomorphism bbp subgraphs bbp maximal extended molecular graphs typically annotated atom bond types preserved isomorphisms general allow weight function weight isomorphism sum weights vertices edges mapped common subgraph isomorphism maximum weight maximum maximum isomorphism map vertices edges contributing weight call pairs forbidden define biconnected mcis outerplanar graphs section present algorithm determine weight maximum common biconnected induced subgraph isomorphism two biconnected outerplanar graphs first show compute maximal common biconnected subgraph isomorphisms since may contain forbidden vertex edge pairs describe obtain weight maximum solution finally show output one maximum solutions outerplanar graphs several characteristic properties see information particular algorithm exploits fact droschinsky kriege mutzel biconnected outerplanar graphs unique outerplanar embedding plane mirror image embeddings every edge incident exactly two faces uniquely defined observe mapping determined starting parameters edge input graphs together mapping endpoints incident faces say face mapped isomorphism vertices bordering face mapped distinguish four cases describe mapping edge edge isomorphism biconnected induced subgraphs assume edge incident faces incident see fig least one face incident must mapped since common subgraph must biconnected sake simplicity case distinction also associate two faces regardless whether mapped isomorphism may map endpoints edges two different two incident faces distinguish following four cases given isomorphism biconnected common induced subgraphs maps two endpoints edge let function type determine type mapping following result key obtain efficient algorithm lemma let maximal isomorphisms biconnected common induced subgraphs biconnected outerplanar graphs assume mapped edge type type proof obvious direction correct prove implication since common subgraph required biconnected isomorphisms must map least one face incident edge face incident two faces well mapping endpoints two edges uniquely determined type mapping consider mapping vertices cyclic border faces since mapping endpoints fixed mapping vertices border face unambiguously determined since common subgraph required biconnected every extension mapping must include vertices neighboring face face mapping endpoints shared edge implicates mapping vertices cyclic border extension unambiguous therefore mapping successively extended unmapped face consequently holds dom dom since maximal possible one extended hence must dom dom result follows proof lemma constructively shows obtain maximal solution given two edges type parameter assume approach realized procedure maximaliso finding largest common substructures molecules quadratic time algorithm outerplanar graphs input biconnected outerplanar graphs output weight maximum common biconnected subgraph isomorphism data table storing weight mapping type forall type valid undefined maximaliso splitiso forall edges mapped mapped split iso type otherwise return maximum entry returns unique maximal isomorphism maps two given edges according specified type algorithm implemented means tree structure encodes neighboring relation inner faces weak dual graphs similar approach running time compute maximal solution note edge pairs four types mappings possible type valid pair edges least one incident face mapped according type edges incident faces bordered number vertices maximal solution may map vertex edge pairs forbidden according weight function order obtain maximum weight split split isomorphisms weight isomorphism biconnected induced common subgraphs split isomorphisms obtained time follows consider graph dom every forbidden edge incident two inner faces split graph connected component graphs delete forbidden vertices edges determine blocks restricted vertices block yields split isomorphism approach realized function splitiso used following every edge mapped one resulting isomorphisms referred every split isomorphism obtained maximal solution algorithm uses table storing weight constraint maps according type size table algorithm starts pairs edges valid types mappings maximal isomorphism biconnected common induced subgraphs computed extending initial mapping splitting maximal solution multiple valid isomorphisms droschinsky kriege mutzel weight obtained weights stored pairs edges contained considering type mapping includes weights occurring forbidden vertices edges keeping values allows avoid generating isomorphism multiple times main procedure loops pairs edges four possible mappings pair note mapping split isomorphisms computed time improved analysis gives following result theorem algorithm computes weight biconnected outerplanar graphs time proof allocate costs call maximaliso followed splitiso cells table mapping containing edges computed time result exactly cells table filled value value cell computed line assures edge mapping specific type used initial mapping corresponding cell already filled every initial mapping extended must lead isomorphism containing edge mappings associated undefined cells according lemma therefore total costs algorithm allocated cells cell pays constant amount proves total running time bounded size table easily modify algorithm enumerate maximum isomorphisms without affecting total running time first run algorithm obtain maximum weight wmax run modified version algorithm outputs every split isomorphism size wmax soon found right splitiso called line solving outerplanar graphs previous section presented algorithm compute two biconnected outerplanar graphs section generalize compute two outerplanar graphs following assume isomorphisms bbp require input graphs connected otherwise compute pairs connected components select isomorphism maximum weight proceed follows first give insight data structure helps partition set bbp common subgraph isomorphisms subsets certain conditions compute isomorphism maximum weight subsets using dynamic programming approach similar one used solve maximum common subtree problem among computed isomorphisms output one maximum weight thus data structure given observe bridges mapped bridges edges one block finding largest common substructures molecules quadratic time input graph fig biconnected outerplanar graph edge incident faces connected outerplanar graph block nodes gray background bridge nodes filled solid black nodes cutvertices corresponding subgraphs shown block bridge nodes mapped edges contained exactly one block mapped edges form biconnected common subgraph connected graph let denote set cutvertices blg set blocks brg set bridges blg brg bcg tree nodes edges nodes iff refer vertices distinguish block nodes bridge nodes example graph bcg shown fig graph define connected component includes least one vertex allow sets component unambiguous example fig graph partitioning bbp isomorphisms first define let arbitrary block bridge define contain isomorphisms least one edge mapped isomorphisms dom defined contain isomorphisms exactly one vertex mapped isomorphism observe disjoint isomorphisms contain vertices let blocks bridges share cutvertex iff node cbi edges bcg isomorphism maps vertex maps vertices one node dom connected definition every recursively define sets isomorphisms described map vertices vbi example consider fig let consist isomorphisms map least one edge edge isomorphisms map exactly one vertex recursion continues three additional sets consist isomorphisms map least one edge three exactly one vertex vertex operating vbi recursion continues two additional sets sets empty droschinsky kriege mutzel partitioning pxy computing isomorphism maximum weight set partition subsets focus separation graph distinguish two cases set least one edge certain block bridge mapped partitioned subsets meaning blh brh mapped vertices mapped terms bbp block bridge preserving intended set exactly one vertex mapped subsets defined follows operate define subset restriction computing maximum isomorphism subset pxy describe compute isomorphism maximum weight subset pxy idea recursively extend mappings vertices two single bridges two single blocks along pairs mapped cutvertices bnodes determined mwms preserving bridges blocks computed isomorphisms select one maximum weight first let pxy subset least one edge mapped edge bridges two possible mappings considered blocks maximal common biconnected subgraph isomorphisms blocks considered alg may given fixed mapping call considered isomorphism valid respects possible fixed mapping contains vertices operating extend valid isomorphisms along pairs mapped cutvertices follows let bcbi path path pair recursively calculate following restrictions cutvertices must mapped bridges blocks iii least one vertex block bridge must mapped restriction iii assures least one vertex added isomorphism therefore recursion compute method described paragraph computing pair construct weighted bipartite graph vertices pair mapped cutvertices weight edge determined weight restrictions subtracted appropriate cutvertices restricted edge computing mwm bipartite graphs determines extension matching edge corresponding computed isomorphisms merged extending valid isomorphisms select one maximum weight second let pxy subset exactly one vertex mapped let cutvertex possible expansion within allowed subset therefore subset contains exactly one isomorphism next assume cutvertex cutvertex may finding largest common substructures molecules quadratic time extend similar previous paragraph difference defined containing reason mapped vertices yet therefore may expand directions cutvertex contained exactly one interested bbp isomorphisms means vertices mapped must block bridge therefore path type compute isomorphism fixed mapping least one edge mapped falls back method paragraph well among computed isomorphisms select one maximum weight appendix lists pseudocode computing described time complexity time compute essentially depends time compute biconnected isomorphisms blocks time compute mwms time compute linear number edges vertices considering pairs blocks theorem bound time computing biconnected isomorphisms need compute mwms pairs cutvertices two graphs follows result theorem maximum common subtree problem total time min log max maximum degree bcg proves following theorem theorem two outerplanar graphs solved time iff biconnected bounded degree otherwise min log max experimental evaluation section evaluate algorithm experimentally compare approach algorithms implemented compiled gcc application running times measured intel core cpu using single core available memory sufficient computations interested answering following questions extent differs molecular graphs large difference terms running time molecular graphs running time affected specific properties input graphs answer extracted randomly chosen pairs outerplanar molecular graphs large chemical molecules grateful leander schietgat providing implementation used nci open database http droschinsky kriege mutzel fig running times computations black dot represents computation two randomly chosen outerplanar molecular graphs directly compares running time algorithm mcis implementation mces running times another bbpmcis computations fit borders database contain vertices vertices average weight function set pair vertices edges label otherwise matches setting answer compared algorithms randomly generated connected outerplanar graphs graph generator takes several parameters input evaluated three different properties graph size average ratio edges vertices average block size outerplanar graphs ratio edges vertices less evaluating effect one property preserved two procedure allows verify whether theoretical findings consistent running times observed practice set weight function pair vertices edges corresponds uniformly labeled graphs comparing weight isomorphisms computed two algorithms observed difference tested molecule pairs suggests yields valid notion similarity outerplanar molecular graphs shown algorithm computed solutions average times faster dots fig represent computation times two algorithms results summarized table schietgat compared finding largest common substructures molecules quadratic time table upper half running times implementation mcis implementation mces lower half relative differences computation times algorithm mcis mces average time median time less maximum time comparison average factor median factor minimum factor maximum factor mces mcis table average time computations random outerplanar graphs varying one property graph size ratio edges vertices block size note units measurement time exceeds days size mcis mces timeout mcis mces mcis mces min timeout algorithm algorithm general mcis algorithm similar computation times small graphs much faster large graphs maximum time general mcis algorithm hours contrast computation time never exceeded clearly indicates algorithm orders magnitude faster general approach first varied size input graphs preserving average ratio edges vertices average block size based theorem expected average time increase factor bit double size results table closely match expectation next evaluated different ratios edges vertices graph size set average block size higher ratio results higher number faces blocks consequently affects time required alg particular table size thus running time expected show quadratic growth increase running time exceeds expectation might explained increasing size data structure used represent faces blocks finally evaluated different average block sizes graph size set average ratio edges vertices higher block sizes mean less mwms compute costly part computation therefore expected running time decrease results shown table support droschinsky kriege mutzel conclusion developed algorithm computes chemical meaningful largest common substructure outerplanar molecular graphs fraction second hence method makes comparison large molecular datasets possible future work would like extend approach general graph classes focus efficiency practice references akutsu polynomial time algorithm finding largest common subgraph almost trees bounded degree ieice transactions fundamentals electronics communications computer sciences akutsu tamura algorithm computing maximum common connected edge subgraph outerplanar graphs bounded degree algorithms droschinsky kriege mutzel faster algorithms maximum common subtree isomorphism problem mfcs lipics vol ehrlich rarey maximum common subgraph isomorphism algorithms applications molecular science review wiley interdisciplinary reviews computational molecular science garey johnson computers intractability guide theory freeman kriege kurpicz mutzel maximum common subgraph problems graphs jan miller froncek eds iwoca lncs vol springer kriege mutzel finding maximum common biconnected subgraphs seriesparallel graphs dietzfelbinger eds mfcs lncs vol springer lingas subgraph isomorphism biconnected outerplanar graphs cubic time theoretical computer science matula subtree isomorphism alspach miller eds algorithmic aspects combinatorics annals discrete mathematics vol elsevier nicholson tsai johnson naim subgraph isomorphism theorem molecular graphs graph theory topology chemistry stud phys theoret elsevier raymond willett maximum common subgraph isomorphism algorithms matching chemical structures journal molecular design schietgat ramon bruynooghe maximum common subgraph algorithm outerplanar graphs application chemoinformatics annals mathematics artificial intelligence syslo subgraph isomorphism problem outerplanar graphs theoretical computer science yamaguchi aoki mamitsuka finding maximum common subgraph partial graph polynomially bounded number spanning trees inf process lett finding largest common substructures molecules quadratic time pseudocode algorithm outerplanar graphs input connected outerplanar graphs weight function output data bcg bch node sets select arbitrary block bridge setsx initial recursion call output procedure setsx input excluded vertices output isomorphism maximum weight initialize empty mapping forall bridges blocks pxy least one edge mapped forall pairs pxy single vertex forall paths bcg setsx vertex mapped recursion return droschinsky kriege mutzel procedure input mapping optional output maximum isomorphism least one edge mapped restricted given exactly one block return block bridge preserving forall valid isomorphisms forall pairs cutvertices mapped forall pairs bcbi paths compute mwm bipartite graph edge weights forall edges extend return procedure input excluded vertices mapping output maximum isomorphism single vertex mapped restricted return expansion possible forall pairs bvbi path compute mwm bipartite graph edge weights forall edges extend else unambiguous node set contains vertex forall bvbi path return
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national laboratory scientific computing feb graduate program computational modeling managing scientific hypotheses uncertain probabilistic data bernardo brasil february managing scientific hypotheses uncertain probabilistic data bernardo thesis submitted examining committee partial fulfillment requirements degree doctor sciences computational modeling approved fabio porto chair pedro dias marco casanova ana carolina salgado brasil february bernardo nunes rights reserved bernardo managing scientific hypotheses uncertain probabilistic data bernardo national laboratory scientific computing xvii orientador fabio porto thesis national laboratory scientific computing hypothesis management predictive analytics uncertain probabilistic data causal reasoning probabilistic database design porto fabio iii cdd dedicatory parents tania francisco special love marcelle island certainty uncertain world acknowledgments thesis work supported lncc graduate program computational modeling cnpq grant faperj grant nota ibm fellowship would like express gratitude advisor fabio porto gift challenging topic thesis special grateful inspiring advisor nicely influencing towards database research also thank thesis committee attention time devoted assessment work indebted ana maria moura generous advice support throughout research frederico silva adolfo support research project would like thank much researchers lncc give lectures graduate program specially karam filho teaching generously roots mathematical modeling contributed significantly education thinker shaping scientific mathematical skills thank colleagues friends lncc specially eduardo lima ramon costa klaus wehmuth raquel lopes karine diego paredes companionship joy shared pursuit theses would also like gratefully recall earlier professors ufes giancarlo guizzardi rosane caruso berilhes garcia shaping essential building blocks education finally would like thank family support father francisco example simplicity goodness mother tania tenacity true love leo sche dear brother sister true union life grandmothers zeca orizontina rosita love prayers thank also marcel maris greatest gift daughter near future wife marcelle best partner one could ever hope originally experimental science theoretical science kepler laws newton laws motion maxwell equations many problems theoretical models grew complicated solve analytically people start simulating simulations carried much last half last century point simulations generating whole lot data along huge increase data experimental jim gray abstract thesis presented partial fulfillment requirements degree doctor sciences managing scientific hypotheses uncertain probabilistic data bernardo february advisor fabio porto view paradigm shift makes science ever thesis propose synthesis method encoding managing deterministic scientific hypotheses uncertain probabilistic data form mathematical equations hypotheses symmetrically relate aspects studied phenomena computing predictions however deterministic hypotheses abstracted functions build upon simon notion structural equations order efficiently extract causal ordering variables implicit hypothesis structure set mathematical equations show process hypothesis predictive structure effectively original algorithms encoding set functional dependencies performing causal reasoning terms acyclic reasoning reasoning reveals important causal dependencies implicit hypothesis predictive data guide synthesis probabilistic database like field graphical models probabilistic database normalized uncertainty arisen competing hypotheses decomposed factors propagated properly onto predictive data recovering joint probability distribution lossless join motivated principle hypothesis management predictive analytics method applicable quantitative qualitative deterministic hypotheses demonstrated realistic use cases computational science resumo tese apresentada como parte dos requisitos para grau doutor como dados incertos bernardo fevereiro orientador fabio porto tendo vista paradigma que faz cada vez mais guiada por dados nesta tese propomos para larga escala como dados incertos forma relacionam simetricamente aspectos estudo para entanto podem ser como levamos adiante simon estruturais para extrair forma eficiente chamada causal estrutura uma mostramos como processar estrutura preditiva uma algoritmos originais para sua como conjunto funcionais realizamos causal termos sobre tal revela importantes causais nos dados preditivos que conduzem nossa banco dados como modelos banco dados deve ser normalizado tal forma que incerteza oriunda alternativas seja decomposta fatores propagada propriamente recuperando sua probabilidade conjunta via isso motivado como projeto para proposto quantitativas qualitativas demonstrado casos computacional table contents introduction problem space specific goals thesis statement thesis contributions thesis outline vision hypotheses data running example hypothesis encoding reasoning uncertainty introduction predictive analytics related work summary key points hypothesis encoding preliminaries structural equations problem causal ordering total causal mappings encoding scheme experiments related work summary results causal reasoning preliminaries armstrong inference rules table contents acyclic reasoning equivalence causal ordering experiments related work summary results probabilistic database synthesis preliminaries probabilistic wsa running example properties related work summary results applicability physiome project testbed case studies system prototype experiments discussion conclusions conclusions revisiting research questions significance limitations open problems future work final considerations table contents bibliography appendix detailed proofs proofs hypothesis encoding proofs causal reasoning proofs probabilistic synthesis list figures figure view deterministic scientific hypothesis view scientific method life cycle usual data ingesture pipeline simulation data management pipeline processing hypotheses uncertain probabilistic data scientific hypotheses seen alternative functions predict data predictive analytics hypothesis evaluation study descriptive textual data example big fact table loaded simulation raw data explanation relational table result set query simulation trial dataset hypothesis predictive tables rendered query using analytics predicted position conditioned observation directed causal graphs associated two systems running simon causal ordering algorithm coa directed causal graph induced mapping structure bipartite graph structure example another hypothesis structure example complete matching structure example encoded set alg structure example performance hypothesis encoding logscale set encoding alg structure fig folding derived alg list figures xiii set encoding structure fig folding performance acyclic causal reasoning logscale generated operation sets encoded given structures hypotheses example big fact table hypothesis example loaded trial datasets identified special attribute tid big fact table hypothesis example emphasized resp colors green red set compare folding projections rendered hypothesis rendered hypothesis plot hemoglobin oxygen saturation hypotheses descriptive textual data example set hypothesis set hypothesis set hypothesis result set hypothesis management query results analytical study hemoglobin phenomenon plot baroreflex hypothesis dahl rat descriptive textual data example result set hypothesis management query results analytical study baroreflex phenomenon set hypothesis plot myogenic behavior hypothesis descriptive textual data example set hypothesis set hypothesis result set hypothesis management query results analytics vessel myogenic behavior phenomenon list figures xiv population observed population observed screenshots first prototype system descriptive textual data example set hypothesis set hypothesis set hypothesis results analytics population phenomenon results analytics hudson bay lynx population phenomenon performance behavior physiome testbed scenario physiome hypotheses used experiments example boolean network hypothesis example boolean network model list tables table simulation data management hypothesis data management acronyms functional dependency probabilistic database probabilistic world set algebra maybms database management system artificial intelligence graphical models bayesian networks etl extract transform load olap analytical processing data warehouse sem structural equation model coa causal ordering algorithm structure hypothesis set equations structure set variables appearing equations structure ars set variables appearing equation total causal mapping equations variables structure set causal dependencies causal graph induced big fact table hypothesis set hypothesis big fact tables synthesized hypothesis set synthesized hypothesis relational explanation table list tables explanation table phenomenon identifier hypothesis identifier sets subset closure set derived closure set folding set synthesis synthesis uncertainty tid hypothesis trial identifier bcnf normal form mathml mathematical markup language tcm total causal mapping algorithm lhs rhs side side sch data columns table condition columns table xvii chapter introduction view paradigm shift makes science ever thesis demonstrate large deterministic scientific hypotheses effectively encoded managed kind uncertain probabilistic data deterministic hypotheses formed principles ideas expressed mathematically implemented program run give decisive form data see fig hypotheses also learned large scale exhibited eureqa project examples structured deterministic hypotheses include tentative mathematical models physics engineering economical sciences conjectured boolean networks molecular biology social sciences important reasoning devices solved generate valuable predictive data decision making science increasingly business well fact refer nowadays broad modern context data science big data complexity scale problems require proper data management tools predicted data analyzed effectively thesis pay attention quite general class tentative computational science look original way distinguished kind data source computational science sic rapidly growing multidisciplinary field uses advanced computing capabilities understand solve complex problems may refer tentative computational science models throughout text structured deterministic law free fall body falls rest velocity point proportional time end iii fall figure view deterministic scientific hypothesis generally considered computational science models interpreted hypotheses explain phenomena strategic relevance usually complex may hundreds thousands intertwined coupled variables computed along space time frequency domains arbitrarily large scale important note distinction structure data levels consider say model essentially consists eqs two ordinary differential equations complemented seven subsidiary equations set values domain variable input parameters sense said fairly simple characterized set equations set variables sized yet data level model chapter made large computing predictions fine time resolution along extended time window shall see shortly technical challenges associated thesis involve majorly structure level model abstracted deterministic structure structure length measure dense hypothesis structure comprising total sum number variables appearing equation really concerned models whose structure order whose results data shall difficult analyze handicrafted practice note data level model set large wanted set domain resolution extension accordingly shall necessarily large structure large hypotheses mean tentative deterministic models large structure level overall class hypotheses said qualify least four five associated notion big value role advancing science technology volume due large scale modern scientific problems variety structural heterogeneity even refer phenomena veracity due uncertainty idea managing hypotheses data may sound intriguing fact raises number research questions conceptual technical start outlining conceptual research questions define encode hypotheses data sources uncertainty may present considered hypotheses data relate observational data likewise phenomena data database perspective every piece simulated data qualify scientific hypothesis difference managing simulation data managing hypotheses data available proper data format use automatically extract hypotheses challenge thesis provide reasonable answers questions brought together vision hypotheses data call vision use case present chapter experiment realistic scenarios chapter velocity may appear connection machine learning hypotheses discuss chapter shall keep record questions revisit problem space specific goals vision formulates problem hypothesis encoding problem probabilistic database design number technical questions arise introduce technical context materials methods identified selected thesis basis realize vision terms probabilistic database design shall outline sequel technical research questions answered core thesis problem space specific goals goal thesis investigate capabilities probabilistic databases enable hypothesis data management particular case simulation data management sequel first characterize use case hypothesis data management formulate terms probabilistic design simulation data management simulation laboratories provide scientists engineers large possibly huge datasets reconstruct phenomena interest high resolution notorious examples john hopkins turbulance databases human brain project hbp neuroscience simulation datasets core motivation delivery data enabling new insights discoveries hypothesis testing observations nonetheless use case exploratory analytics currently well understood many challenges already coped simulation data increasingly accessible recently part thesis work use case hypothesis management taken account predictive analytics fact pressing call innovative technology integrate observed data simulated theories unified framework point raised leading neuroscientists context hbp incisive compelling argument massive simulation databases constrained experimental data corrective loops test precise hypotheses fig shows simplified view scientific method life cycle distinguishes phases exploratory analytics context problem space specific goals context discovery context justification phenomenon observation hypothesis formulation computational simulation testing data valid yes publishing results figure view scientific method life cycle highlights hypothesis formulation backward transition reformulation predictions disagree observations discovery predictive analytics context justification highlights loop hypothesis formulation testing simulation data generated tuned combination theoretical empirical principles distinctive feature considered compared data generated technology scientific experiments pronounced uncertainty component motivates use case hypothesis data management predictive analytics essential aspects hypothesis data management described contrast simulation data management follows table summarizes comparison sample data hypothesis management shall deal volume data simulation data management exploratory analytics samples aligned example architectural design cern experiment simulation atlas four data volume data significantly decreases raw data data actually used analyses hypothesis testing samples raw table simulation data management hypothesis data management simulation data management exploratory analytics raw data extremely large access pattern denormalized faster retrieval data updates hypothesis data management predictive analytics sample data large access pattern normalized uncertainty factors probability distribution updates problem space specific goals tion data selected comparative studies involving competing hypotheses presence evidence sample observational data principle also aligned data delivered model repositiories since observations usually less available fragment sample simulation data matches coordinates sample observations required simulation results comparative analysis instance show predictive analytical study extracted virtual physiological rat project comparing sample simulation data heart rates baroreflex model observations dahl rat simulation originally set produce predictions time resolution since observational sample fine gain rendering predicted sample hypothesis testing note sampling incur additional uncertainty typical statistical sampling access pattern simulation data management access pattern based selected coordinates data denormalized faster retrieval typical data warehouses olap particular account big table approach state modeled physical system recorded large single row data fairly reasonable etl data ingesture pipeline characterized updates see fig setting fact fit exploratory analytics entire states simulated system shall accessed providing data visualization system altogether data retrieval critical risk update anomalies hypothesis management contrast centered claims identified within hypothesis structure available data dependencies since focus resolving uncertainty decision making http analytical processing distinguished oltp transaction processing latter meant transaction processing daily queries updates operational systems former analytical queries data warehouses gather lot data collected different sources decision making problem space specific goals etl sim figure usual data ingesture pipeline simulation data management datasets generated byssimulation trials hypothesis models loaded big table uncertainty buried database lacks logical organization enabling hypothesis management predictive analytics hypothesis best fit data must normalized based uncertainty factors key correctness uncertainty modeling efficiency probabilistic reasoning say probabilistic database uncertainty modeling uncertain probabilistic data management uncertainty may come two sources incompleteness missing data multiplicity inconsistent data hypothesis management sample simulation data concerned multiplicity prediction records due competing hypotheses targeted studied phenomenon multiplicity naturally gives rise probability distribution may initially uniform eventually conditioned observations conditioning applied bayesian inference problem translates database update transforming prior probability distribution posterior overall hypothesis data management also yet markedly different simulation data management key point distinguishes hypothesis management fact unit data defined predictive content every predicted fact data dependencies claim accordingly data decomposed organized access pattern problem space specific goals conditioning etl dkp figure pipeline processing hypotheses uncertain probabilistic data hypothesis structure given spin format sample simulation data trials dki indicated target phenomenon say loaded big table snthe synthesis comes play read base possibly many hypotheses transform probabilistic database hypothesis decomsm posed claim tables probability distribution computed phenomenon covering hypotheses trials targeted distribution updated posterior presence observational data anticipate chapter synthesis method developed thesis work processing hypotheses uncertain probabilistic data comprises pipeline see fig extends one shown fig probabilistic database design probabilistic databases evolved mature technology last decade emergence new data models query processing techniques one probabilistic data models representation system probabilistic algebra implemented maybms elegant extension relational model shall refer thesis management uncertain probabilistic data look point view design formal design methodology yet proposed despite advanced state probabilistic data management techniques lack methods systematic design may prevent wider adoption availability design methods considered one key success factors rapid growth applications field graphical models considered inform research analogously proposed distinguish methods design three classes subjective construction learning problem space specific goals data iii synthesis kind formal specification first less systematic user model data correlations steering construction process maybms use cases illustrated way second comprises analytical techniques extract data learn correlations external sources possibly unstructured schema prevalent one date motivated information extraction data integration applications thesis present methodology third kind extract data dependencies previously existing formal specification hypothesis mathematical structure synthesize algorithmically type construction method successful building bayesian networks knowledge thesis first synthesis method design shall develop means extract specification hypothesis encode hypothesis management analytics shall flatten deterministic hypotheses synthesis method developed relies extraction functional dependencies basic input algorithmic example consider relation fall fig holds meaning values attribute time functionally determine values attributes velocity position precisely let two tuples rows instance relation table fall satisfies iff implies illustrative relation fall particular key constraint means values play role key provide access values relation related concept also major one normalization ensure resulting design process bears desirable properties associated notion normal form hypothesis management uncertainty modeled normalized uncertainty one claim may undesirably mixed fact considered critical failure traditional design lack techniques obtain important information real world problem space specific goals uncertainty another claim expected involve processing causal dependencies implicit given hypothesis structure shall introduce detail concepts context necessary structural equations flattening user mathematical models hypothesis nonetheless straightforward goal thesis investigate proper abstractions mathematical models order partly capture semantics extent tailored hypothesis management opposed say model solving shall abstract mathematical models intermediary artifacts amenable encoded fact given system equations set variables appearing seminal article simon introduced asymmetrical functional relation among variables establishes causal ordering became known structural equation models sem structural equations also along lines goal extract causal ordering implicit structure deterministic hypothesis set guides synthesis shall see throughout text causal ordering capture process provides causal dependencies implicit predictive data useful information decompose uncertainty sake probabilistic modeling reasoning uncertainty model uncertain probabilistic data management essentially two sources uncertainty incompleteness missing data multiplicity inconsistent data kind uncertainty dealt work multiplicity hypothesis trial records identified targeted phenomenon record uncertainty arises existance competing hypotheses multiple hypotheses trials inserted phenomenon system interprets defining probability distribution problem space specific goals probability distribution usually uniform multiplicity competing hypotheses accordance probability theory semantics modeled data model operators implemented maybms system shall see conf aggregate operator instance spite name performs standard probabilistic inference probability distribution eventually however need condition initial probability distribution presence observations conditioning shall adopt bayesian inference prior probability distribution updated posterior informal discussion section opens way number technical research questions outline next algorithm given sem efficiently extract causal ordering computational properties problem connection sem devise encoding scheme orient equations effectively transform one guarantees properties set set ready used schema synthesis encoding hypothesis causal structure kind processing perform efficiently reasoning directly relate sem causal ordering uncertainty decomposition required predictive analytics reducible structure level processing need process simulated data identify additional uncertainty factors finally properties desirable schema targeted hypothesis management ensured synthesis method given machinery process hypotheses relational properties detect hypotheses back system hypothesis management delivered top maybms backend thesis statement conceptual level technical means speak hypotheses good terms principles philosophy science core thesis devoted answer questions shall accomplish throughout chapters thesis statement statement thesis possible effectively encode manage large deterministic scientific hypotheses uncertain probabilistic data key challenges conceptual technical nature conceptually provide core abstractions define encode hypotheses data technically provide number algorithms compose designtheoretic pipeline encode hypotheses uncertain probabilistic data verify efficiency correctness applicability effectiveness method demonstrated realistic case studies computational science besides worthwhile highlighting thesis although perform sort information extraction acquisition hypotheses model repositories web basic order obtain testbed method proposing means systematic extraction hypotheses available sources fact shall outline important direction future work address solving computational models numerical analytics sense fact rely numerical solvers implemented tools use transaction processing systems load computed data relational big fact table render tables synthesized method deal data visualization either sense efficiency scalability query processing particular urelational maybms rely addressed thesis contributions evaluated thesis fact performance extensively evaluated shown effective performance tests carried thesis comprise techniques encoding synthesis hypothesis databases terms uncertainty statistical analysis stick process forms multiplicity data constitute model uncertainty dealt work relying maybms perform probabilistic inference iii eventually application level perform bayesian inference posterior probability distribution propagated updates provide additional form uncertainty management rather manage data extracted system user control process uncertainty terms specific sources uncertainty recognized chapter thesis contributions contributions thesis outlined follows innovative contributions thesis presents vision hypotheses data use case socalled vision published vision track vldb sic potentially visionary content innovative system described system prototype demonstration paper technical contributions thesis presents specific technical developments vision short shows encode deterministic hypotheses uncertain probabilistic data detailed technical contributions chapters formulated formal method design hypothesis described technical report method together realistic testbed scenarios performance evaluation yet published preliminary version available corr preliminary version available corr thesis outline thesis outline structure remainder thesis outlined reference chapter vision research vision hypotheses uncertain probabilistic data characterization use case key points technical challenges presented chapter encoding problem encoding hypothesis data given formal specification set mathematical equations presented addressed encoding scheme transforms equations guarantees terms preserving hypothesis causal structure chapter causal reasoning presented technique causal reasonig acyclic reasoning encoded processes hypothesis causal ordering find first causes predictive variables chapter synthesis presented technique address problem uncertainty introduction propagation transformation hypotheses databases synthesized shown bear desirable properties hypothesis management predictive analytics chapter applicability discussion applicability implementation proposed techniques prototype system test demonstration vision realization realistic case studies presented chapter conclusions research questions revisited significance limitations thesis directions future work final considerations discussed chapter vision hypotheses data technology scientific experiments provide scientists empirical data extracted transformed loaded ready analysis vision consider theoretical data data generated simulation deterministic scientific hypotheses also needs analyzed hypotheses data view age science consider deterministic scientific hypotheses point view formed principles learned large hypotheses formulated mathematically coded program run give decisive form data see fig uncertain data semantic structure relation fall fig item expressed functional dependency typical semantics assigned empirical data design experiment databases dimension like time example used key observables like velocity position empirical uncertainty physical dimension keys like may violated say alternative sensor readings hypotheses however tentative explanations phenomena characterizes different kind uncertain data order manage theoretical uncertainty shall need two special attributes compose say epistemological dimension keys observables identifying studied phenomena identifying hypotheses aimed explaining shall leverage semantics relations like fall leap core abstraction exhibited eureqa project given state predicted state figure deterministic scientific hypotheses seen alternative functions predict data giving rise theoretical empirical sources uncertainty vision predictive data scientific hypotheses tested way predictions form mathematical equations hypotheses symmetrically relate aspects studied phenomenon however computing predictions deterministic hypotheses applied asymmetrically functions take given valuation input variables parameters produce values output variables predictions observing shall seek principled method transform symmetric mathematical equations hypothesis asymmetric looking deterministic hypotheses alternative functions predict data see fig vision shall deal two sources uncertainty given context set alternative hypotheses aimed explaining providing predictions selected phenomenon theoretical comprises selecting best tentative model function produce best data empirical comprises candidate model parameter input setting calibrates best way selected phenomenon note two sources uncertainty intertwined one clean one without cleaning neither theory parameters directly multiplicity hypothesis entries associated phenomenon multiplicity hypothesis trial entries associated phenomenon observable joint results predictions thesis aim providing means support kind integrated analytics applications big computational science research programs human brain cardiovascular applications challenged theoretical big data users need analyze results hundreds thousands simulation trials besides recent initiatives model repositories fostering model integration sharing reproducibility computational sciences growing reasonably fast web promoting standard model specification limited integrity lack support competing models two reasons provide strong use case vision hypothesis management physiome project planned integrate several large deterministic models human physiology fairly simple model human cardiovascular system variables also pressing call deep predictive analytic tools support users assessing scenarios business enterprises deep predictive analytics based first principles deterministic hypotheses beyond descriptive analytics shallow predictive analytics statistical forecasting ibid ratifies hypothesis management promising class applications probabilistic vision currently set delivered top probabilistic algebra developed influential maybms implied design principles compositionality ability introduce uncertainty maybms query language fits well hypothesis management shall look previously mentioned point view synthesis method design shall particularly make use repair key operation gives http http concept deep predictive analytics haas discussed detail project website http maybms backend extension postgresql offers traditional querying capabilities latter addition uncertain probabilistic running example figure predictive analytics hypothesis evaluation study hypotheses simulated data compete explain phenomenon observed data rise alternative worlds repairs argument key predictive analytics tool database research uncertainty usually seen undesirable property hinders data quality shall refer implemented maybms nonetheless show ability introduce controlled uncertainty otherwise complete simulation dataset tool deep predictive analytics set competing alternative hypotheses shows scenario hypotheses data compete explain phenomenon roadmap remainder chapter claim hypotheses encoded identified see uncertainty quantified probability distribution see browsed user selectivity criteria furthermore probabilities conditioned possibly presence evidence see running example let consider example presentation vision example research conducted effects gravity falling object earth atmosphere scientists uncertain precise object running example density predominant state fluid solid three hypotheses considered alternative explanations fall see fig due parameter uncertainty six simulation trials run four phenomenon description effects gravity object falling earth atmosphere hypothesis name law free fall stokes law law figure descriptive textual data example tid figure big fact table hypothesis loaded simulation raw data trials identified tid construction data warehouse requires simple user description research descriptive records phenomena hypotheses dimensions see fig inserted first basic referential constraints satisfied associated datasets fact tables instance one six trial datasets hypothesis shall reference foreign key table hypothesis synthesized relations fig shows big fact table hypothesis loaded trial datasets phenomenon although table denormalized faster data retrieval usual extraction hypothesis equations allows render automatically since variables must appear equation proceed hypothesis encoding start address research questions hypothesis encoding hypothesis encoding aim extracting hypothesis set mathematical equations suppose given set equations hypothesis let examine set target law free fall order derive equations focus implicit data dependencies get rid constants possibly complex mathematical constructs equation written way roughly speaking suggests prediction variable functionally dependent physical dimension parameters yet dependency like may hold infinitely many fact need way identify mathematical formulation precisely abstraction semantics achieved introducing hypothesis special attribute see data representation deterministic scientific hypothesis built encoding scheme see leverages semantics structural equations special attribute phenomenon supposed key values parameters determination parameters empirical phenomenondependent task expectedly violated user recall rigorous presentation method encode set due chapter think say many polynomials satisfy dependency reasoning uncertain values parameters rationale applies derive equations vary structure include parameter object diameter stokes law law key point hypothesis structure set equations given format mathematics method extract hypothesis set equations carefully designed based hypothesis data representation abstraction fact shall explore mathml format hypothesis reasoning hypothesis set extracted reasoning performed discover implicit data dependencies fact dependency theory equipped formal system reasoning sets like derive closure elaborate chapter shall particularly concerned inference rule applied instance gives inference allows observe factor uncertainty dimensional key constraint values fact note derived like constraint values expectedly violated presence uncertainty http uncertainty introduction serve fig multiplicity values pair functionally determine reason admit special attribute trial tid overimposed trivial repair provisionally uncertainty introduced controlled way synthesis meant identify simulation trials pretend certainty lose integrity data imposed certainty raw simulation trial data safely loaded files see fig note however certainty held expense redundancy mostly important opaqueness predictive analytics since tid isolates hides inconsistency violated constraints next stage construction pipeline uncertainty introduced controlled manner uncertainty introduction proceed uncertainty introduction procedure note relation fig predicted acceleration values association hypothesis target phenomenon established fact insertion hypothesis trial dataset user must set target phenomenon may quite convenient design decision envisioned system hypotheses abstract universal statements derived predictions empirically grounded assigning callibrating onto phenomenon assignment set data entry time fact holds data recorded explanation table named default see fig top provided weights establishing prior probability distribution user choice may may uniform data transformation certain uncertain relations starts query whose result set materialized table see fig introduce detail schema set pairs condition columns map discrete random variable created operation one possible values hypotheses universal definition must qualify class different situated phenomena predictive datasets must specific one specific situation uncertainty introduction conf figure explanation relational table associated table rendered application operation world table internal maybms automatically stores marginal probabilities formal semantics operation given create table select repair key weight conf semantics seen generalization data cleaning context data cleaning rather carried gradually keeping mutually inconsistent tuples probability distribution ibid updated face evidence probabilities tuples eventually tend zero eliminated motivates remark remark consider table fig note abstracts goal hypothesis evaluation study scientific method repair key users develop research directly upon data support query update capabilities hypotheses relationship repaired function phenomenon best explanation given big fact table need correlated input attributes independent uncertainty units one associated random illustrate means query materializes view let identified attribute inferred input parameter means reasoning uncertainty introduction tid tid figure result set query simulation trial dataset hypothesis create table select repair key select count group weight result set stored see fig note possible values mapped random variable table considered source joint probability distribution values input parameters may uniform count frequency possible value done pass argument construct far presented informally procedure proceed presented rigorously chapter consider order synthesize predictive since functionally determined independent propagate uncertainty onto query create table select query shown selects result sets resp shown fig predictive analytics figure predictive tables rendered query using compare relations accounting correlations captured could propagate onto uncertainty coming hypothesis parameter sensible thus precisely situating tuples space possible worlds done predictive attributes end shall ready predictive analytics competing predictions possible alternatives mutually inconsistent key point synthesis process amenable algorithm design except user research description construction fully automated based hypothesis structure set equations raw hypothesis trial data predictive analytics users example able say query phenomenon predicted position specific values time considering hypotheses admitted illustrated query creates integrative table query computes confidence aggregate operation tuples fig shows result apart column posterior confidence hypothesis specific prediction split due parameter uncertainty sum back total confidence since parameter uncertainty factors certain possible values shown possible tuples considering hypotheses phenomenon confidence predictive analytics values sum one accordance laws probability create table select select union select union select select conf prior group order prior desc prior posterior figure analytics predicted position conditioned observation users make informed decisions light confidence aggregates eventually conditioned face evidence observed data example features kind bayesian conditioning discrete random variables mapped possible values predictive attributes like position whose domain continuous example suppose position feet observed secs standard deviation applying bayes theorem normal mean discrete prior prior updated posterior see fig related work procedure uses normal density function say get likelihood alternative prediction mean given observed applies bayes rule get posterior general case examples shown chapter actually phenomenon data sample independent observed values brazil population observed census years likelihood competing trial computed product single likelihoods bayes rule settled compute posterior given prior result prior probability distribution assigned via repair key eventually conditioned observed data applied bayesian inference problem translates update one induce effects posteriors back table first prototype system accomplish performing bayesian inference application level applying update variant sql update maybms solution good enough let complete use case demonstrations related work vision managing hypotheses data roots porto spaccapietra motivated conceptual data model support socalled silico science means scientific model management system chapter particular related work discuss work understand mostly related vision datadriven hypothesis management analytics haas provide original perspective evolution database technology characterize data typically managed traditional systems record past conclusion insight solution context scientific databases position suggestive technology designed empirical data theoretical data generated simulation principles scientific hypotheses recognize current technology raised art scalable descriptive analytics high level point however nowadays sic enterprises really need prescriptive analytics identify optimal business policy investment engineering decisions face uncertainty analytics turn shall rest deep predictive analytics beyond mere statistical forecasting imbued understanding fundamental mechanisms govern system behavior allowing analyses sum pressing call deep predictive analytic tools business enterprises much science comparison vision haas proposing research program pursue data management technology deep predictive analytics discuss strategies extend query engines model execution within along lines query optimization understood general problem connections algebraic solvers framework turn essentially comprises abstraction technique encoding hypotheses data understood comparison putting models strictly flattened data perspective reason directly applicable building upon recent work principle integrated say olap layer project scientific simulation data previsouly mentioned science etl distinguished unfrequent updates large raw files data sources challenges related work enabling efficient access raw simulation data documented supercomputing database research viewpoints pointed key use case exploratory analytics extreme scale raw data motivated approaches data exploration immersive query processing move program data situ query processing raw files exploit spatial structure data indexing schemes line research motivated equipping scientist immediate interaction large simulation nodb approach particular argues eliminate etl phase loading direct access data situ raw data files fact data exploration fundamental use case science nonetheless generated first principles learned deterministic hypotheses simulation data pronounced uncertainty component motivates another use case case hypothesis management predictive analytics motivated latter requires probabilistic design enabling uncertainty decomposition factorization hypothesis management shall deal volume data simulation data management exploratory analytics samples table comparison instance cern experiment atlas four data management volume data significantly decreases raw data data actually used analyses hypothesis testing overall overhead incurred loading samples raw simulation trial datasets justified enabling principled hypothesis evaluation according scientific method hypothesis encoding framework comparable bioinformatics initiatives address hypothesis encoding rdf data model robot scientist system kbs automated generation testing hypotheses sometimes phrased files queries results summary key points genes encode enzymes yeast organism hybrow kbs scientists test hypotheses events galactose metabolism also yeast organism iii swan kbs scientists share hypotheses possible causes alzheimer disease robot scientist relies logic programming analytics automatically generate test hypotheses kind gene function empirical data hybrow likewise hypotheses formulated user biological events swan turn disfavors analytic techniques hypothesis evaluation focus descriptive aspects hypotheses natural language statements retrieved publications hypothesis associated claims rdfencoded meant support basis empirical evidence data particular swan differs former hypothesis unstructured related efforts retrieval textual claims narrative fabric scientific reports though consist rdf encoding sequence genome analysis hypotheses varying levels structure gene function statements free text framework turn consists encoding hypotheses mathematical equations knowledge first work hypothesis relational encoding finally hypothesis evaluation comparison analytics vision distinguished terms bayesian inference approach latter pointed major direction improvement bioinformatics initiatives mentioned fact influential model decision making hypothesis evaluation summary key points outline key points vision structured deterministic hypotheses encoded theoretical data distinguished empirical data introduction epistemological dimension semantic structure summary key points two sources uncertainty considered theoretical uncertainty originating competing hypotheses empirical uncertainty derived alternative simulation trials hypothesis phenomenon method extract structure hypothesis carefully designed based hypothesis data representation shall reducible terms format mathematical modeling mathml shall adopt standard hypothesis specification seen controlled introduction uncertainty simulation data amenable algorithm design reducible designtheoretic synthesis method construction simulation data modeled hypothesis data whenever associated target phenomenon phenomenon may happen associated many hypotheses research activity modeled data cleaning problem essentially vision comprises pipeline fig insertion hypothesis shall given structure together simulation trial datasets raw files apply etl automatic procedure generate hypothesis big fact table trial extracted equations firstly encoded time many hypotheses may inserted system uncertainty introduction procedure applied process encoded synthesize uncertain eventually conditioned observations note fig etl procedure operated local view hypothesis procedure conditioning operated global view available hypotheses pipeline opens four main tracks technical research challenges etl stage hypothesis encoding causal reasoning iii synthesis conditioning address sequel three first track challenges depth problem conditioning outlined work chapter hypothesis encoding chapter address problem hypothesis encoding introduce notation basic concepts structural equations problem causal ordering study problem extracting causal ordering implicit structure deterministic hypothesis show simon classical approach intractable build upon less notorious approach nayak borrow efficient algorithm fits well use case hypothesis encoding develop encoding scheme builds upon idea structural equations original abstraction hypotheses present experiments attest encoding scheme works practice large hypotheses discuss related work summarize results chapter preliminaries structural equations given system mathematical equations involving set variables build structural equation model sem essentially establish mapping equations variables shall enable detecting hidden asymmetry variables causal ordering instance einstein famous equation states equivalence mass energy summarizing theory imputed two different asymmetries different applications say given fixed amount mass recall constant predict particle relativistic rest energy given particle rest energy predict mass potential nuclear fission preliminaries structural equations figure directed causal graphs associated two systems stress point consider newton second law scalar setting modeler either use compute predict say acceleration values given amount mass different force intensities predict force intensities given fixed acceleration testing engineered dynamometer point newton equation enough derive predictions number variables larger must completed two equations order qualify applied hypothesis although usually interpreted asymmetry towards technically nothing semantics suggest compare two systems given fig sum causal ordering system equations guessed inferred chapter rely previous work mostly work adapt encoding hypotheses def structure pair set equations set variables subset equations structure least different variables appear subset equations variables appear values variables chosen arbitrarily values equality construct used predicate assignment operator shall introduce notion directed causal graphs shortly preliminaries structural equations remaining variables determined uniquely finding unique values matter solving equations def let structure say complete short interested systems equations structural def complete def many equations variables subset equations fewer variables complete structures solved unique sets values variables work however concerned solving sets mathematical equations processing causal ordering view design simon concept causal ordering roots econometrics studies taken flavor graphical models gms thesis translate problem causal ordering language data dependencies def let structure say minimal complete complete structure def structure matrix structure matrix entry aij variable appears equation zero otherwise elementary row operations row multiplication constant structure matrix may hinder structure causal ordering valid general also emphasizes problem causal ordering solving system mathematical equations structure identifying hidden asymmetries def let complete structure total causal mapping bijection ars also expect systems equations given input independent sense linear algebra context means systems equations case subset equations fewer variables equations system must preliminaries structural equations structure matrix given coa execution recursive steps figure running simon causal ordering algorithm coa given structure fig minimal subsets detected recursive step highlighted different shades gray diagonal elements colored fig simon informally described algorithm given complete structure used compute partial causal mapping partitions set equations partitions set variables shown dash druzdzel causal mapping returned simon socalled causal ordering algorithm coa total variables strongly coupled determined simultaneously also shown total mapping must consistent coa partial mapping latter made partial design merge strongly coupled variables partitions clusters order force induced causal graph acyclic algorithm coat variant simon coa adapted illustrate use case returns total causal mapping instead partial causal mapping illustrate example fig example consider structure whose matrix shown fig note complete since minimal set minimal subsets eliminating variables identified recursive step smaller structure derived compare partial causal mapping eventually returned coa total causal mapping returned coat problem causal ordering algorithm coat variant simon coa procedure coat structure require given complete ensure returns total causal mapping minimal store minimal structures found return coat return since strongly coupled see coat maps arbitrarily could instead total mapping renders cycle directed causal graph induced see figure directed causal graph induced mapping structure edge connects node towards node iff appears equation problem causal ordering serious issue alg coa finding minimal structures given structure line hard problem addressed heuristically problem also called biclustering problem causal ordering boolean matrices simon approach however shall see next way cope problem causal ordering fact order study computational properties sem problem causal ordering observe structure satisfying def modeled straightforwardly bipartite graph set equations set variables disjoint vertex sets edge set connecting equations variables appearing fig shows bipartite graph corresponding structure given example comprehensive text graph concepts related algorithmic problems even figure bipartite graph structure example biclique complete bipartite graph bipartite graph every two vertices note balanced bicliques degree deg vertex must deg recent approaches problems come notion also called relaxation biclique concept allow less rigid notion connectivity complete connectivity required biclique recall simon coa needs find recursive step minimal subsets theorem situates particular computational task terms complexity equivalent find recursive step problem causal ordering least corresponding bipartite graph see fig take def specific notion def let bipartite graph say vertices deg state originally balanced problem bpbp decision problem follows bpbp given bipartite graph positive integer contain lemma balanced problem bpbp proof show restriction bpbp generalization balanced biclique problem bbp referred balanced complete bipartite subgraph problem shown means transformation clique restriction bpbp bbp special case made requiring def either deg equivalent ways enforcing inquired biclique introduce another hypothesis structure see fig illustrate correspondence property coa algorithmic approach elaborated proof theorem theorem let complete structure extraction causal ordering simon coa proof show recursive step coa find minimal subsets translates optimization problem associated decision problem bpbp know lemma see appendix note clearly positive integer total causal mappings coa execution recursive steps bipartite graph figure another hypothesis structure example nonetheless problem causal ordering solved efficiently means different less notorious approach due nayak introduce build upon next total causal mappings problem causal ordering solved polynomial time finding total causal mapping structure given def computing transitive closure set direct causal dependencies induced exists ars def let structure variables total causal mapping inducing set direct causal dependencies transitive closure say direct causal dependency causal dependency words iff direct causally depends given causal asymmetries induced notions open approach causal reasoning fits well use case aimed encoding hypothesis structures sets performing symbolic causal reasoning terms acyclic reasoning chapter effective nonetheless shall need ensure properties total causal mappings first note differs research geared reasoning total causal mappings given structure may multiple total causal mappings recall example causal ordering must unique see fig therefore question arises whether transitive closure total causal mapping proposition originally nayak ensures case proposition let structure two total causal mappings proof proof based argument nayak present arguably much clearer way see appendix intuitively shows differ variable equation mapped variables must causally dependent strongly coupled another issue concerned precise conditions total causal mappings exist whether variables equations causally determined fact proposition based nayak apud hall know existence condition holds iff given structure complete proceeding let refer even briefly introduce additional concepts necessary matching graph subset edges two edges matching share common node matching said maximum edge added matching without hindering matching property finally matching graph said perfect every vertex edge matching bipartite graph perfect matching said complete matching proposition let structure total causal mapping exists iff complete proof observe total causal mapping corresponds exactly complete matching bipartite graph fact even apud hall theorem know complete matching iff every subset vertices total causal mappings set vertices connected vertices edges def subset equations fewer variables equations def number equations number variables easy see conditions hold iff complete structure problem finding maximum matching algorithmic problem thesis adopt algorithm known bounded handle problem total causal mapping see alg translating problem maximum matching bipartite graph linear time applying algorithm get matching finally translate back total causal mapping suggested proof proposition algorithm find total causal mapping given structure procedure tcm structure require given complete structure ensure returns total causal mapping translates structure bipartite graph solves maximum matching problem translates matching total causal mapping return fig shows complete matching found algorithm structure given example algorithm solves maximum matching bipartite graph efficiently problem finding maximum flow network total causal mappings figure complete matching structure example corollary summarizes results far corollary let complete structure total causal mapping found alg tcm time bounded proof let bipartite graph corresponding complete structure given tcm translation done scan scan length note number edges rendered precisely length structure denser structure greater matching computed internal procedure turn done expense thus easy see tcm dominated maximum matching algorithm known since assumed complete therefore tcm must running time remark let complete structure know proposition total causal mapping exists let defined tcm causal ordering implicit shall correctly extracted proposition processing causal dependencies induced show chapter ready accomplish hypothesis encoding show next encoding scheme encoding scheme shall encode variables relational attributes map equations onto total causal mappings let set attribute symbols complete structure let two special attribute symbols kept identify resp phenomena hypotheses explicitly distinguishing symbols assigned user structure epistemological symbols consider sense simon nature scientific modeling interventions summarized def def let structure variable say exogenous exists equation ars case written must mapped total causal mapping say endogenous otherwise remark introduces interpretation def data dependency flavor remark values exogenous variables attributes determined empirically outside system proposed structure values therefore dependent phenomenon values endogenous variables attributes turn determined theoretically within system dependent hypothesis shall dependent phenomenon well indirectly introduced encoding scheme presenting obvious goes beyond simon structural equations abstract semantics mathematical deterministic hypotheses whereas simon structural equations able represent linear equations encoding scheme represent equations arbitrarily complex mathematical operators means data representation deterministic hypotheses instance take equation suppose considering context complete system equations alg tcm maps onto variable abstraction equation semantics shall encode encoding scheme figure encoded set alg structure example hypothesis identifier captures semantics hypothesis encode complete structures sets means alg fig presents set defined encoding structure example algorithm hypothesis encoding procedure structure domain variables require given complete structure ensure returns set tcm exogenous supress dimensions like time else endogenous return study properties encoded sets shall make use concept canonical sets also called minimal see def def let set say canonical note without hypothesis infinitely many equations fit pattern experiments form set satisfying properties def individually say said attribute extraneous def finally said trivial note presence trivial set sufficient make redundant theorem let set defined complete structure may may canonical proof show properties def must hold produced alg property may hold encoded set may see appendix draw attention significance theorem sheds light connection simon complete structures sets fact continue elaborate connection next chapter handle causal ordering processing symbolically causal reasoning experiments fig shows results experiments carried order study effective procedure hypothesis encoding practice particular behavior hypotheses whose structure randomly generated orders magnitude length largest structure considered generated exactly related work executed ten runs tested order magnitude taken mean running time plot shown fig logscale base fact slope expected curve structure length time scalability results compatible computational complexity corollary bounded time structure length figure performance hypothesis encoding logscale related work modeling physical systems set equations traditional modeling approach large bulk models exist date simon early work structural equations causal ordering comprises specific notion causality aimed contributing potential modeling approach meant identifying influences among variables values implicit system model enabling informed interventions may apply either system phenomenon study model say predictions approximating observations well significant research effort devoted causal modeling reasoning past decades statistics notion causality used traced back early work simon others econometrics experiments performed intel core running mac note arbitrary structure worst case densest structure possible establish time bound function related work nonetheless two important differences emphasized work majorly devoted deal statistical qualitative hypotheses deterministic quantitative hypotheses causal model assumed given derived data instead converted synthesized set equations core differences also apply work comparison bulk existing work probabilistic main point though clarify technical context state art problem causal ordering works concerned extracting causal model previous existing formal specification set equations reason causal ordering yet barely studied problem computational point view dash druzdzel revisit problem light modern applications first provide formal description simon coa gives summary causal dependencies implicit given sem clustering strongly coupled variables causal graph coa provides condensed representation causal model implicit given sem show valid total causal mapping produced given sem must consistent coa partial causal mapping yet serious problem algorithm turns intractable fact formal study coa computational properties yet found literature thesis obtained negative hardness result intractable turns compatible nayak intuition sic exponential time algorithm inspired serrano gossard work constraint modeling reasoning nayak reports approach provably quite effective process causal ordering extract total causal mapping compute transitive closure direct causal dependencies thesis build upon perform causal reasoning terms form transitive reasoning approach fits well use case synthesis show summary results chapter process causal ordering hypothesis structure abstracted sem terms acyclic causal reasoning prove correctness enabled encoding scheme presented chapter summary results chapter studied developed encoding scheme process mathematical structure deterministic hypothesis set towards encoding hypotheses studied properties held encoded set list results achieved follows theorem know original hardness result simon approach process causal ordering structure intractable building upon work simon nayak propositions framed approach efficiently extract basic information total causal mapping processing causal ordering implicit mathematical structure deterministic hypothesis corollary know process complete structure hypothp esis total causal mapping time bounded machinery hypothesis encoding provably suitable large hypothesis structures theorem studies properties encoded sets unraveled connection simon complete structures sets explore next chapter performed experiments fig study effective approach practice scales hypotheses whose structure randomly generated length order tests order hardware limitations experimental settings theory complexity time bounds larger structures handled efficiently chapter causal reasoning chapter present technique address problem causal ordering processing order enable synthesis introduce armstrong classical inference system reason develop core concept algorithm folding set method acyclic causal reasoning show connections equivalence causal reasoning present experiments method behaves practice discuss related work conclude chapter preliminaries armstrong inference rules usual notational conventions literature write denote sets relational attributes denote singleton attribute sets also write shorthand functional dependency theory relies armstrong inference rules axioms reflexivity augmentation transitivity forms sound complete inference system reasoning one derive additional rules decomposition union acyclic reasoning given set one obtain closure finite application rules concerned reasoning set order process implicit causal ordering latter shall see performed terms transitive reasoning note particular case shall refer reasoning understand included next definition opens way compute efficiently let set attributes attribute closure set attributes bernstein long given algorithm alg xclosure compute polynomial time resp given set attribute set defined tighter time bound linear time achievable discussed remark algorithm attribute closure procedure xclosure set attribute set require set attribute set ensure attribute closure size size size return consumes acyclic reasoning discussed previous chapter shall process causal ordering terms computing transitive closure endogenous variable predictive attribute proceed shall develop machinery reason terms armstrong rule shall acyclic reasoning demonstrate correspondence kind reasoning causal reasoning shortly sequel def let set attributes closure minimal set iff derived finite possibly empty application rule case may write omit understood context fact interested specific proper subset say kernel gives compact representation causal ordering implicit note characterize special subset shall need careful presence cycles causal ordering def let set attributes say folded write intuition def folded sense going reasoning anymore nothing new discovered given set shall able find folded applying much possible ruling cyclic trivial clever way def let set attributes attribute folding attribute set accordingly folding proper subset iff example continued fig shows set left folding right note folding obtained computing attribute folding acyclic reasoning figure set encoding alg structure fig folding derived alg illustrate reasoning steps partially compute attribute folding considering subset given given given note still amenable application say derive however even though resp form well characterizes cycle fetches nothing fact consider given satisfies def folded holds empty application lemma let complete structure total causal mapping set encoded given attribute folding exists unique proof see appendix note step deriving cycle yet formed acyclic reasoning give original algorithm alg compute folding set core lies alg afolding understood variant xclosure alg designed acyclic reasoning order compute folding attribute algorithm afolding seen backtracing causal ordering implicit towards analogously terms directed graph induced causal ordering see fig would comprise graph traversal identify nodes reachability rather afolding processing causal ordering fully symbolic based armstrong rewrite rule example cyclicity set may effect making folding degenerate instance consider note canonical afolding given given algorithm folding set procedure folding set require given encodes complete structure ensure returns set folding afolding return acyclic reasoning algorithm folding attribute set procedure afolding set attribute require parsimonious ensure returns attribute folding consumed consumed attrs stores attrs found causally dependent size size halts size consumes consumes attr cyclic reingests simulate cyclic app return theorem let complete structure set encoded given let afolding correctly computes attribute folding time proof proof roadmap note afolding monotone size increases terminates precisely denotes attributes step outer loop folding step shall prove induction given attribute parsimonious returned afolding unique attribute folding see appendix remark let arbitrary set attribute set beeri bernstein gave straightforward optimization alg xclosure make linear maximum length string encoding note actual length string case exactly equivalence causal ordering optimization mentioned applies likewise alg afolding implemented run linear time corollary let complete structure set encoded given algorithm folding correctly computes folding time time complexity alg afolding proof see appendix finally shall convenient come notion parsimonious sets see def suggestive distinguishing feature mathematical information systems comparison arbitrary information systems def let set attributes say parsimonious canonical proposition shall useful connection concept folding proposition let complete structure total causal mapping set encoded given let folding parsimonious proof see appendix equivalence causal ordering show equivalence acyclic reasoning causal ordering processing start theorem establishes equivalence notion causal dependency encoding scheme presented chapter omit tedious exposure short shall require one auxiliary data structure keep track yet consumed many attributes yet consumed appear rhs equivalence causal ordering theorem let complete structure total causal mapping set encoded given causally dependent iff proof prove statement induction consider first direction converse see appendix def gives useful terminology neat concept towards goal chapter def let structure variables total causal mapping inducing set direct causal dependencies transitive closure say first cause proposition connects notion first cause exogenous endogenous variables introduced chapter proposition let structure variable first cause exogenous accordingly variable first cause endogenous proof straightforward definitions see appendix note exogenous variables encoded since values variables assigned outside system remark devoid indirect causal dependencies uncertainty except thus concerned processing causal uncertainty chaining towards goal rather find first causes endogenous variables predictive attributes shall need terminology def introduce lemma paving way goal def let structure set encoded subset accordingly subset equivalence causal ordering figure set encoding structure fig folding fig illustrates set folding applied subset order compute first causes endogenous variables lemma let complete structure total causal mapping set encoded given variable first cause variable either proof prove statement construction theorem see appendix finally theorem clarifies purpose folding meaning terms causal ordering theorem let complete structure total causal mapping set encoded given let attribute encodes variable every first cause encoded attribute proof show existance missing first cause folded leads contradiction see appendix note folding taken endogenous variable experiments remark observe one hand goal computing transitive closure set induced causal dependencies derive entire causal ordering given structure goal folding hand discover variables attributes given variable attribute causally dependent first causes particular results shown comprise method compute endogenous variable predictive attribute first causes core goal reasoning device developed chapter order enable automatic synthesis hypotheses uncertain probabilistic data experiments fig shows results experiments carried order study effective causal reasoning practice particular behavior hypotheses whose structure randomly generated orders magnitude length largest structure considered generated exactly like experiments previous chapter executed ten runs tested order magnitude taken mean running time plot shown fig logscale base notice linear rate growth across orders magnitude base sized structures growth factor structure length doubled time required causal reasoning grows factor doubled well scalability results compatible computational complexity folding bounded yet bit overestimated time bound see plot fig related work concept set folding design alg afolding quite obvious variant xclosure original approach problem processing causal ordering hypothesis via acyclic reasoning best knowledge specific form reasoning experiments performed intel core running mac related work time folding structure length figure performance acyclic causal reasoning logscale yet unexplored problem database research literature reasoning extensively covered maier recent years seen emergence foundational work causality databases motivated improving usability terms providing users explanations query answers essentially idea borrowed work causality identify causal ordering tuples given query result set system able explain user tuples caused answer possibly expected tuples missing requires causal chain tuples given query computationally expensive database instance large conjunctive queries causality said computed efficiently specific problem addressed kanagal sensitivity analysis aimed establishing refined connection query answer output elements instance input supporting user interventions instead providing user causes goal enable user know changes input affect output line work strongly related vision reverse data management causal reasoning presence constraints yet unexplored topic though called worth future work meliou rich information exploited sake explanation sensitivity analysis available intuitive search space problems shall significantly reduced summary results fact encoding equations captures causal chain exogenous input endogenous output tuples schema level nonetheless form causal reasoning geared hypothesis management analytics uncertainty management point view concrete connection causality yet established summary results chapter studied developed technique acyclic causal reasoning list results achieved follows developed principled concepts core algorithm alg folding order perform acyclic reasoning towards efficient method causal reasoning yet elegant database formalism systematic construction hypothesis probabilistic given reasonably tight time bound behavior reasoning device terms structure given input established theorem corollary time bound folding algorithm shown correctness folding algorithm connection causal reasoning theorem theorem defined core notion first causes def proposition meant guide procedure chapter precisely capturing uncertainty factors endogenous variables predictive attributes similar yet markedly different computing transitive closure causal dependencies remark performed experiments fig study effective approach causal reasoning practice scales hypotheses whose structure randomly generated length order experiments show time bound though already effective large structures bit overestimated chapter probabilistic database synthesis chapter present technique synthesize hypothesis stage pipeline relational schema loaded datasets computed hypotheses alternative trials input settings challenge model design probabilistic version render suitable hypothesis management analytics introduce present running example illustrate uncertainty introduction procedure pipeline fig present technique factorize uncertainty present big fact table terms uncertainty factors show propagate uncertainty predictive attributes properly based first causes detected shown chapter discuss related work finally conclude chapter preliminaries probabilistic wsa three remarkable features expressiveness closed positive relational algebra queries succinctness efficient storage large number possible worlds vertical decompositions support attributelevel uncertainty efficient query processing including confidence computation database finite set structures postpone presentation experiments synthesis whole preliminaries probabilistic wsa relations numbers possible world probability element probabilistic algebra consists operations relational algebra operation computing tuple confidence conf repairkey operation introducing uncertainty giving rise alternative worlds repairs argument key let relation possible world let contain numerical values greater zero let satisfy maximal repair fig schema set pairs condition columns map discrete random variable one possible values world table stores marginal probabilities notion illustration data transformation certain uncertain relations consider query extension relational algebra whose result set materialized shown fig conf also let sch sch two union pairs condition columns operations selection projection product issued relational algebra rewritten positive relational algebra sch sch sch consistent running example conf figure generated operation pairs condition columns returns pairs classical relations rewrite rules apply accordingly rewriting parsimonious translation sic number algebraic operations increase operations selection projection remains kind query plans hardly complicated input queries fact verified hat relational database query optimizers well practice comprehensive overview refer reader thesis look point view design methodology yet proposed concerned particular hypothesis management applications running example proceeding consider example fairly representative illustrate deal correlations predictive data deterministic hypotheses sake suitable analytics example explore three slightly different theoretical models population dynamics applications ecology epidemics economics etc malthus model logistic equation model practice equations meant extracted xml files chapter consider ordinary differential equation notation running example read variable function time given initial condition models completed user additional equations provide values exogenous variables input parameters sem resp fig shows sets encoded structures given also consider trial datasets hypothesis model loaded big fact table relation shown fig admit special attribute trial tid keep hypothesis trials identified uncertainty introduced controlled way synthesis stage fig given actually task encoding algorithm tcm infer variable whether exogenous endogenous means total causal mapping recall domain variables like time informed encoding algorithm suppress running example figure sets encoded given structures hypotheses example tid figure big fact table hypothesis example loaded trial datasets identified special attribute tid given big fact table synthesis two main parts process empirical uncertainty present big fact table synthesize decompose independent propagate precisely predictive data seen operation allows one create discrete random variable order repair argument key given relation goal devise technique perform operation principled way hypothesis management basic design principle exactly one random variable distinct uncertainty factor short requires carefully identifying actual sources uncertainty present relations multiplicity competing hypotheses standard one theoretical consider explanation table like fig stores foreign keys hypotheses available target phenomena take explanation table three hypotheses example discrete random variable defined query formula considered standard synthesis repair key standard hypotheses nonetheless abstract universal statements order produce concrete valuation endogenous attributes predictions one inquire particular situated phenomenon tentatively assign valuation exogenous attributes eventually tuned target multiplicity competing empirical estimations hypothesis leads problem learning empirical problem let set encoded given hypothesis structure big fact table relation loaded trial data let set attributes encoding exogenous variables problem learning infer casual strong input correlations form maximal groups attributes casual hold pick group pivot representative insert set tid figure big fact table hypothesis example emphasized resp colors green red learning meant process attributes inferred exogenous given hypothesis latter attributes officially unrelated fact casual mean correlations set experimental trials may occasionally show trial input data hold related principle theory fig helps illustrate problem big fact table emphasize colors green red observe values strongly correlated values like note also seen certain factor user point view record reflects common practice computational science known parameter sensibility analysis problem dominated problem discovery relation really new problem see keep focus synthesis method whole omit detailed algorithm figure set compare folding output set filled completed projection illustration consider hypothesis trial input data recorded fig show resulting set fig left together folding right latter input alg merge get final information necessary actual synthesis captured def illustration merging equivalent sides note fig right holds def let complete structure big fact table hypothesis set defined let folding finally define merge say algorithm merge equivalent sides procedure merge set holds sci merges equivalent keys else return short make use relational algebra operation build pruned lattice attribute groups number rows grouping similarly remark let structure big fact table every encodes claim either empirical theoretical ensured merge algorithm groups equivalent sides able employ notion decomposition formulated def query formula extension relational algebra def let complete structure hypothesis big fact table let set attributes exists define yki query formula say yki projection yki count count relational algebra grouping operator figure projections rendered hypothesis synthesis projections particular application important consequence introduction new defined follows see def shall consider rather study properties synthesized def let resp complete structure big fact table hypothesis let indexes projections say repaired factorization central part pipeline recall machinery developed far hypothesis encoding causal reasoning enabling predictive analytics let briefly reconstruct hypothesis structure take endogenous variable encoded attribute exactly one theorem every first cause observe rendered learning filled partly partly processed summarize exogenous variables independent means learning first cause encoded shall represented pivot attribute occasionally strongly correlated hold subject folding processing contains first causes contains pivot representatives meant enabling economical representation uncertainty running example small principle quite relevant hypotheses say correctness summarization shall ensured proposition let know parsimonious def canonical therefore def form almost ready upropagation note result pivot attribute associated random variable yki shall use surrogate yki order propagate factorized uncertainty predictive ykj join formula attribute sets defined merging equivalent lhs get pass argument synthesis let contains domain variables pivot attributes shall included data columns ykj leave trace condition columns annotate sch ykj repair key def abstracted general query formula employed alg synthesize accomplish part def let complete structure hypothesis big fact table let define ykj query formula say ykj predictive projection ykj yki yki projection take set pivot attributes representing first causes algorithm synthesis applied folding set procedure synthesize structure big table explan table require complete ensure returned bcnf lossless decomposition merge folding part scans hypothesis count count yki part scans claims hypothesis prepares keep track pivot attributes first cause indexes projection keeps track pivot attribute removes pivot attributes ykj return fig shows rendered hypothesis whose big fact table shown fig note tid corresponds defines particular world whose probability value derived marginal probabilities stored world table see fig result application formulas remark observe although alg synthesize operates locally hypothesis effects synthesis pipeline global account global explanation relation see fig fact probability tuple row say ykj hypothesis distributed among hypotheses keyed hypotheses compete figure rendered hypothesis properties rendered synthesis ready querying typical queries comprise conf aggregate operation inquiring probability confidence tuple true probability space captured hypothesis competition illustrate queries chapter properties procedure meaningful study properties projections prediction projections synthesized big fact table sake predictive analytics particular projections submit satisfy normal form bcnf def repaired factorization correct decomposition uncertainty present join lossless preserve data def note study consider repaired factorization since one actually holds key repairing decomposition emphasized remark every claim remark holds repaired factorization thus claimcentered decomposition big fact table desirable schema satisfies bcnf bcnf represent fact twice notion good design uncertainty decomposition view predictive analytics avoid uncertainty one claim undesirably mixed uncertainty another claim def let relation scheme set attributes set projection onto written subset def let relation scheme set attributes set say bcnf superkey properties schema bcnf schemes bcnf example illustrate concept bcnf let consider canonical set attributes tentative schema containing single relation abc relation bcnf one violates superkey observe also overdecomposed schema may trivially satisfy bcnf example let def schemas abc bcnf second however breaks data two tables making access difficult necessary since brings information schema synthesized would desirable apply union merge point target schema also desirable schema bcnf theorem guarantees bcnf property design every schema rendered alg synthesize theorem let resp complete structure big fact table hypothesis let repaired factorization explanation table hypothesis recorded let schema defined synthesize bcnf proof exploit fact projection onto projections predictive projections define disjoint partition since know form search space bcnf violations significantly reduced minimality turn comes alg merge see appendix correctness uncertainty decomposition recall preliminaries equivalent relational product operation main join operation introduced provide classical definition lossless join properties decomposition data relation two relations known preserve data original form application join def let relational schema synthesized collection let set attributes say lossless join every instance satisfying lossless join property interest ensure decomposition data big fact table projections preserves data join annotate predictive projections propagated means join operation correct theorem guarantees case theorem let complete structure hypothesis big fact table repaired factorization explanation table hypothesis recorded let schema defined synthesize join subset projections lossless predictive projection ykj result join theoretical big fact table turn projections yki lossless proof make use lemma ullman proof comes straightforwardly see appendix remark significance theorem lies guarantees decomposition uncertainty based causal ordering processing fact desirable predictive analytics theorem turn significant ensures empirical uncertainty implicit hypothesis big fact table decomposed projections independent strongly correlated problem fully recovered join lossless repaired factorization structure essential make sure composition required recovers uncertainty associated predictive data since repaired factorization related work known correct processing causal ordering results chapter altogether theorem guarantees first causes joined together correctly towards predictive variables influenced seen synthesis technique presented essentially targeted properties also motivated computational performance uncertainty decomposition desirable also speed probabilistic inference fact procedure fully grounded implemented maybms system computational performance dominated query processing present experimental studies procedure designed applicability point view goal provide reference computational measures prospective users related work informed research graphical models suciu provide striking motivation work probabilistic database design design probability distributions large sets random variables decomposed factors simpler probability functions small sets variables factors identified using set axioms graphoids reasoning probabilistic independence variables design principle sic applies data decomposed simplest components key constraints hold table bcnf correlations guide table decomposition simpler tables ideally original table probability distribution recovered query view decomposed tables followed principle decomposition predictive analytics fact connection database normalization theory factor decomposition graphical models discussed verma pearl explored since date formal design theory step direction taken sarma initiative revisits dependency theory view reformulating uncertain schema related work design work takes different direction refer classical dependency theory operations viz operator construct systematically scratch focused extraction processing towards factorized schema synthesized schema ensured bcnf lossless join despite major differences synthesis method builds upon classical theory relational schema design synthesis classical design synthesis criticized due strong uniqueness assumption reduces problem design symbolic reasoning arguably neglecting semantic issues probabilistic design however roots statistical design problem less amenable human factors extract dependencies formal specification design synthesis nothing translating seamlessly reduction made user tentative model studied phenomenon last decade seen significant research effort make systems really usable framework also understood technique design instance comparison crius system supports another kind design approach provides users direct manipulation interface increasingly add structure data dependency extraction processing instead completely alleviates user burden data organization also related probabilistic design topic conditioning firstly addressed koch olteanu motivated data cleaning applications introduced assert operation implement kind knowledge compilation world elimination face constraints fds hypothesis management nonetheless need apply bayes conditioning asserting observed data constraints presented example settles kind conditioning problem relevant vision chapter present realistic use cases addressed problem application level order complete realization vision real prototype system formulation bayes conditioning extension say data model open future work summary results summary results chapter studied developed technique synthesis probabilistic geared predictive analytics completes pipeline fig conditioning performed iteratively algorithm synthesize gives general formulation perform uncertainty introduction causal dependencies given form problem given definition uncertainty factor learning data available given relation remark provides example predictive projections resulting synthesis corresponding probability distributions stored world table remark theorem shown schema synthesized processed causal reasoning bcnf fact decomposition desirable predictive analytics theorem ensures uncertainty decomposition correct original probability distribution big fact table fully recoverable lossless join chapter applicability chapter show applicability scenarios present use cases computational physiology extracted physiome introduce physiome project providing testbed physiome case studies show construction application hypothesis management analytics present prototype system demonstrate running example introduced present experiments physiome hypotheses provide general discussion applicability assumptions scope conclude chapter physiome project testbed physiome project initiative seriously address problems reproducibility model integration sharing computational physiology essentially comprises curated repository computational physiology models available online researchers mathematical modeling language mml allow models written declarative form exported number http physiome model repository expanded models including models extracted sources biomodels cellml archive kegg pathways converted mml automatically case studies interoperable formats environment called jsim allow researchers code mml models straightforwardly run different parameter solver settings build customized data plots see results point view physiome external data source provides interesting testbed realistic scenarios extract physiome models means wrapper implemented read xmml files jsim xml encoding mml models simulation trial datasets rendered parametrized unix script developed invoke jsim automatically batch mode currently physiome wrapper designed read mat files load model input parameter settings data associated model output computed predictive data simulation trial physiome keep records phenomena repository observational data attached entries model repository models appear filter models data meaning one observational datasets plots showing model data fits observations shall make use model entries containing observational data realistic scenarios presented paper case studies section present use cases extracted physiome model case hemoglobin oxygen saturation case stress potential hypothesis analytics comparison handcrafted curve fitting visual analysis study three different hypotheses perform closely visually compared target phenomenon dataset see fig empirically set fit possible observations dataset local view separate compared together global view http http case studies figure plot hemoglobin oxygen saturation hypotheses curves target observations dataset source physiome example resources example shown fig consider physiome model entries described relation hypothesis associated phenomenon described relation phenomenon explanation relation one single hypothesis trial best fit considered hypothesis hypothesis phenomenon name description hill equation binding hemoglobin hemoglobin saturation curve using adair equation hemoglobin saturation curve varied levels description hemoglobin oxygen saturation observational dataset sevenringhaus figure descriptive textual data example ids physiome model repository http case studies encoding encoding hypotheses shown resp fig fig fig delta max min figure set hypothesis delta max min figure set hypothesis hprbc hprbc hprbc hprbc delta max min figure set hypothesis symbol mappings seen insertion hypothesis trial datasets requires users specify target phenomenon corresponding mappings hypothesis symbols target phenomenon symbols use case case studies hypothesis management query illustrates feature hypothesis management case consider user interested predictions subset domain result set shown fig select phi upsilon tid union select phi upsilon tid union select phi upsilon tid order upsilon tid tid figure result set hypothesis management query hypothesis analytics fig shows results analytics conditoning probability distribution presence observations dataset fact hypothesis provides best explanation studied phenomenon enabled application bayesian inference implemented within system contribution methodology equip users tool hypothesis management analytics conf tid prior posterior figure results analytical study hemoglobin phenomenon case studies case baroreflex dysfunction dahl rat case extracted virtual physiological rat show potential hypothesis management analytics model tuning fig shows best fit baroreflex model observational dataset acquired experiment dahl rat turn use carry hypothesis management analytics generate parameter sweep script trials insert database best fit selected automatically bayesian inference figure plot baroreflex hypothesis dahl rat target observations data source example resources example shown fig consider single hypothesis entry described relation hypothesis phenomenon described relation phenomenon parameter sweep trials inserted management analytics hypothesis phenomenon name baroreflex description physiological model full baroreflex heart control system based experimental measurements description baroreflex dysfunction dahl rat figure descriptive textual data example encoding encoding hypothesis shown fig http case studies symbol mappings consider user provides symbol mappings time time hypothesis management query consider user interested time instants heart rate higher threshold say result set shown fig select phi upsilon tid time order time tid tid time figure result set hypothesis management query hypothesis analytics fig shows results analytics phenomenon conditioning probability distribution presence observations dataset since case deals model tuning slightly different parameter settings trial ranking decided small differences posterior probability distribution fig conf tid time prior posterior figure results analytical study baroreflex phenomenon case myogenic behavior blood vessel computational models physiology may account diverse effects take place different levels biological organization organ cellular case studies period beta hrmin hro delta pfast delta pslow gamma delta delta hro delta time delta time min delta tau hro delta delta delta pfast gamma time delta delta pslow time min tau ach delta pslow delta smax delta ach ach delta pmax delta time time min tau ach time min time ach tau ach ach tsmax tsmin alpha cns alpha tpmax tpmin alpha cns alpha gcns alpha cns zeta delta delta eps eps wall eps time min eps eps time min eps wall kne time delta eps eps eps time min eps time min eps eps time min time eps eps eps time min eps time min time eps eps wall time min bwall cwall time hrmax hro delta smax hrmin hro delta pmax time data time min beta bwall ach time min cwall eps time min eps time min eps time min gamma gcns hrmax hrmin hro ach kne time delta time max time min tpmax tpmin tsmax tsmin zeta alpha alpha time min delta pslow time min delta time min delta ach tau ach tau tau ach tau figure set hypothesis molecular levels typically sophisticate model developed incrementally say adding detail previously existing model extending dimensionality extending stationary dynamic account phenomena case study example consider alternative models myogenic behavior reference human blood vessel case studies figure plot myogenic behavior hypothesis according trial target observations diameter example see fig consider physiome model entries displayed relation hypothesis two phenomena see relation phenomenon one trial considered hypothesis two hypothesis hypothesis phenomenon name description myogenic compliant model simulates flow passive vessel actively responding vessel driven sinusoidal pressure input myo dyn resp wfit model describes dynamic response vessel step increase intraluminal pressure description dynamics vessel diameter response pulsatile intraluminal pressure figure descriptive textual data example encoding encoding hypotheses shown resp fig fig symbol mappings consider user provides symbol mappings time diameter time diameter case studies fcomp fout fin pin pout fout min fcomp pin pin ttarget atarget atarget cglobal cmyo pin min ttarget taud min atarget taua ttarget min cglobal cmyo min pmean pamp pmean tnorm pin cglobal cmyo cglobal cmyo pamp pext pmean pout min delta max min taua taud tnorm figure set hypothesis ttarget atarget atarget cglobal cmyo min ttarget taud min atarget taua ttarget min cglobal cmyo min delp cglobal cmyo delp cglobal cmyo delta max min taua taud figure set hypothesis hypothesis management query illustrates feature hypothesis management case user selects diameter predictions within time interval plot fig result set shown fig system prototype select phi upsilon tid union select phi upsilon tid order upsilon tid tid figure result set hypothesis management query hypothesis analytics fig shows results analytics phenomenon conditoning probability distribution presence observations davis sikes myo digdata dataset conf tid time diameter prior posterior figure results analytics vessel myogenic behavior phenomenon case study two tentative models considered uniform prior probability distribution updated posterior distribution note even though hypothesis probability weight concentrated single trial bayesian inference able indicate best explanation tid particular best fit system prototype first prototype system implemented java web application pipeline component server side top maybms backend system prototype extension postgresql developed demonstration prototype whole pipeline fig exploring use case scenarios section provide brief demonstration system population dynamics scenario previously introduced thesis demonstration unfolds three phases first phase show etl process give sense user terms simple phenomena description hypothesis naming file upload get phenomena hypotheses available system managed data second phase reproduce typical queries hypothesis management like shown previous section third phase enter hypothesis analytics module user chooses phenomenon hypothesis evaluation study system lists predictions probabilities selectivity criteria population year predictions ranked according probabilities conditioned observational data available chosen phenomenon demo screenshots fig shows screenshots system fig shows research projects currently available user figs show etl interfaces phenomenon hypothesis data definition synthesis insertion hypothesis trial datasets explanations hypothesis towards target phenomenon fig shows interface basic hypothesis management listing predictions given simulation trial figs show two tabs hypothesis analytics module selection observations viewing corresponding alternative predictions ranked conditioned probabilities demo case population dynamics case refer problem computational science population dynamics scenarios demonstrate system prototype fig shows census data collected fig shows observational data collected hudson bay population https system prototype figure census population figure population observed hudon bay system prototype research dashboard login phenomenon data definition hypothesis data definition hypothesis management analytics selected observations tab analytics ranked predictions tab figure screenshots first prototype system system prototype example see fig consider model entries displayed relation hypothesis two phenomena see relation phenomenon three trials considered hypothesis six hypothesis turn two trials considered hypothesis six trials hypothesis note data definition interfaces figs hypothesis name malthusian growth model description exponential growth model growth population proportional size considered first principle population dynamics logistic equation model introduces growth saturation malthusian model due limitation resources model model describes interactions phenomenon description population lynx population hudson bay canada figure descriptive textual data example encoding encoding hypotheses shown resp fig fig fig see hypothesis structure processing fig min delta max min min figure set hypothesis min delta max min min figure set hypothesis symbol mappings consider user provides following symbol mappings resp phenomena see interface mapping symbols fig system prototype bpt delta max min min min figure set hypothesis year population year population year lynx year lynx year lynx hypothesis management query illustrates feature hypothesis management case user selects hypothesis model filters available data trial tid phenomenon formbased query result set shown fig select order hypothesis analytics fig fig show results analytics resp phenomena conditoning probability distribution presence resp observational datasets first one user verifies hypothesis malthusian model unlikely competitive hypothesis logistic equation approximation population dynamics user knows current trials reasonable trials malthusian model hardly could outperform trials logistic equation studied phenomenon experiments conf tid year population prior posterior figure results analytics population phenomenon conf tid year lynx prior posterior figure results analytics hudson bay lynx population phenomenon see interfaces figs experiments efficiency scalability representation system query algebra extensively demonstrated hypothesis must therefore efficient scalable arbitrary experiments see fig provide measures performance method particular context physiome testbed purpose provide concrete feel efficient methodology however tests four graphs bottom fig involve data level require hardware current experimental setup personal computer allows reach scale experiments performed intel core running mac maybms postgresql extension experiments xml extraction time time encoding time time structure length structure length ntrials ntrials time time conditioning ntrials ntrials figure performance behavior physiome testbed scenario uncertain data processed synthesis sized two first graphs xml extraction encoding collected response time measure interest different structure lengths one corresponds real physiome hypothesis table fig last hypothesis table used tests four last graphs fig conditioning set different number trials ntrials one last test four graphs trials processing uncertain data fits machine main memory interpret performance results shown graphs follows measure performance extraction fluctuation may due practicalities xml dom access methods point performance study practical experiments hypothesis name regulatory vessel myo dyn resp wfit myogenic compliant vessel baroreceptor sarcomere energetics comp four gen weibel lung cardiopulmonarymechanics cardiopulmonmechgasbloodexch highlyinteghuman highlyinthuman wintervention baroreflex figure physiome hypotheses used experiments measures amount time taken process representative hypothesis structures note even structures size amount time required extract hypothesis kept subsecond order magnitude interactive response time personal machine encoding fluctuation expected due varying degrees coupling variables hypothesis structures note although provides good measure size complexity extent intricate cause impact encoding procedure case kept point provide notion amount time required encode representative hypotheses scalability test encoding procedure fig procedure composed learning zation observed previous tests dominated learning component discovery occasional big fact table however longer case implemented workaround keeping addition big table table containing exogenous input parameter variables negligible size data endogenous predictive output variables became subsecond procedure became dominated fact procedure carried discovered also discussion negligible processing time latter expensive synthesis method conditioning conditioning procedure run selected phenomenon composed four main parts first operation conf performs probabilistic inference proper predictive projection big fact table hypothesis associated phenomenon second combines results union query whose result set predictive table third loads phenomenon observation sample data predictive data table memory apply bayesian inference finally prior probability distribution predictive table updated posterior corresponding marginal probabilities updated original tables tests procedure carried varying number trials ntrials total response times shown last plot fig performance behavior interpreted context etl loading setting overhead shall though much lower machines overhead nonetheless justified use case hypothesis management analytics opposed simulation data management exploratory analytics discussion verified hypothesis shown coincide results model tuning described physiome model entries related publications validates applicability methodology tool analysis realistic scenarios current practice computational science model evaluation comparison presence observational data somewhat handcrafted model agreement assessed either qualitatively referring curve shapes data plots quantitatively means scripts methodology offers tool perform hypothesis analytics directly discussion database support querying capabilities therefore potential step towards higher standards reproducibility scalability realistic assumptions core assumption framework hypotheses given formal specification encodable sem complete satisfies defs also semantic assumption standard scientific modeling consider correspondence entities symbols within structure must appear science use cases involving deterministic models assumptions quite reasonable topic future work explore business use cases well hypothesis learning user method hypothesis formation irrelevant framework long resulting hypothesis encodable sem promising use case incorporate machine learning methods framework scale hypotheses evaluate querying capabilities consider learning equations say eureqa qualitative hypotheses methodology primarely motivated computational science usually involving differential equations however applicable qualitative deterministic models well boolean networks consist sets functions boolean expression instance fig presents system boolean equations tentative boolean network model plant hormone fig published notation sphk read like ordinary differential equation next state value variable sphk given state value variable aba parameters kind model variable initial conditions http http discussion sphk aba sphk pld pld phc aba aba atrboh phc ros atrboh ros phc phc ros aba nos nos adprc cadpr adprc cgmp plc aba plc inspk aba inspk cis cgmp cadpr atpase caim cis atpase anionem phc phc depolar kev anionem kout caim ros depolar kout phc ros depolar kap phc depolar kev pepc aba malate pepc aba anionem aba actin closure kout kap anionem actin malate figure example boolean network hypothesis conclusions figure example boolean network model source several kinds dynamical system modeled formalism applications grown gene regulatory network social network stock market predictive analytics even richer semantics considered fuzzy logics encoding method applicable likewise long equations still deterministic conclusions chapter demonstrated discussed applicability methodology referred use case scenarios derived physiome research project shown detail process building representative models physiome model repository qualitative assessment followed experiments provide concrete feel performance behavior models mathematical variables chapter conclusions chapter revisit research questions addressed thesis point significance limitations list open problems topics future work conclude final considerations revisiting research questions let revisit conceptual technical research questions define encode hypotheses data sources uncertainty may present considered chapter provided core abstractions compose vision hypotheses data vision problem hypothesis encoding defined addressed chapter distinguished two main sources uncertainty model uncertainty hypothesis management theoretical uncertainty arising competing hypotheses empirical uncertainty arising alternative trial datasets available hypothesis hypotheses data relate observational data likewise phenomena data database perspective also chapter presented conceptual framework defined hypotheses data shown compared phenomena fact hypothesis management really revisiting research questions significant possible hypotheses presence partial piece evidence every piece simulated data qualify scientific hypothesis difference managing simulation data managing hypotheses data early table provided comparison simulation data management hypothesis data management furthermore scientific research process abstracted chapter problem data cleaning hypotheses seen applied science point view reduced data piece simulation data considered hypothesis whenever assigned explain specific phenomenon available proper data format use automatically extract hypotheses anticipated chapter adoption xml data model general data format extracting hypothesis specifications particular since deal mathematical hypotheses refer mathml standard hypothesis specification concretely chapter present use case demonstration scenarios developed specific wrapper extraction hypotheses specified mml mathematical modeling language algorithm given sem efficiently extract causal ordering computational properties problem shown chapter simon treatment problem causal ordering given sem chapter discussed problem detail presented effective efficient algorithmic approach problem computational cost whole process hypothesis encoding bounded experiments show approach performs well practice large hypotheses revisiting research questions connection sem devise encoding scheme orient equations effectively transform one guarantees properties set also chapter presented algorithmic encoding scheme transform sem set guarantees terms preserving hypothesis causal structure study problem revealed interesting properties resulting sets particular always comparison arbitrary information systems precise economical sense given attribute exacly one rhs set ready used schema synthesis encoding hypothesis causal structure kind processing perform efficiently reasoning directly relate sem causal ordering discuss chapter encoded set must processed find first causes predictive variable addressing chapter presented concept folding set efficient algorithm compute also shown equivalence reasoning causal ordering processing uncertainty decomposition required predictive analytics reducible structure level processing need process simulated data identify additional uncertainty factors finally properties desirable schema targeted hypothesis management ensured synthesis method chapter presented conceptual framework address synthesis uncertainty particular introduced need process hypothesis trial datasets available presented efficient algorithm factorize propagate overall uncertainty present given hypothesis competing explanation target significance limitations phenomenon motivated bcnf notion good design factorized set based folding concept lossless join property required correctness uncertainty decomposition shown synthesized schema bears properties given machinery process hypotheses relational properties detect hypotheses back conceptual level technical means speak hypotheses good terms principles philosophy science equipped machinery proposed thesis able given sem automatically extract causal ordering detect strongly coupled components decide given predictive projection associated projections shall able well query hypothesis ranking phenomenon interest technical means extract hypothesis empirical content predictive power unravel cohesiveness parsimonious terms number different claims epistemological units carried within well empirical grounding first causes finally shall able appraise face competing alternative explanations significance limitations thesis addresses pressing call hypothesis management analytics reasons contribute significance listed structured deterministic hypotheses shown encodable uncertain probabilistic data based principles study connection sem contribution computational properties causal ordering problem causal reasoning open problems future work first synthesis method construction previous existing formal specification definition concrete use case hypothesis management analytics new class applications introduced settled problem bayes conditioning limitations thesis listed bayesian inference implemented application level yet formulated principled technical solution within research encoding scheme transform mathematical system hypothesis set enabling synthesis applicable structured deterministic models stochastic ones open problems future work open problems topics future work listed particular order design dedicated algebraic operation bayes conditioning investigation data dependency formalisms dependencies approximate conditional extend scope towards structured stochastic models development techniques systematic hypothesis extraction welldefined problem web information extraction investigation business use case scenarios decision making top definition machine learning use case scenario industrialize hypothesis formation assess performance feasibility scenario final considerations development automatic data sampling techniques leverage data definition hypotheses phenomena statistical point view final considerations thesis developed vision essentially abstraction hypotheses uncertain probabilistic data comprises methodology systematic construction management hypothesis meant provide principled approach enable scientists engineers manage evaluate scientific hypotheses theoretical data addressed core technical challenges vision order properly encode deterministic hypotheses uncertain probabilistic data envisioned jim gray scientific method shifting towards operated discipline rapidly gaining ground thesis strived proposing core principles techniques enabling hypothesis management analytics opening promising line research probabilistic databases simulation data management bibliography hey tansley tolle fourth paradigm intensive scientific discovery microsoft research schmidt lipson distilling natural laws experimental data science washington dhar data science prediction communications acm jagadish gehrke labrinidis papakonstantinou patel ramakrishnan shahabi big data technical challenges communications acm 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dynamic model guard cell abscisic acid signaling plos biology bibliography fan geerts xiong discovering conditional functional dependencies ieee transactions knowledge data engineering appendix detailed proofs proofs hypothesis encoding proof theorem let complete structure extraction causal ordering simon coa intractable proof show recursive step coa find minimal subsets translates optimization problem associated decision problem bpbp know lemma first recall def structure complete structure given fig left note def minimal structure minimal structures easy see corresponding bipartite graph must number edges vertices must deg accordance def intuition elaborated follows point matter big structure equations ars variables grouped local patterns sparsest kind densest construct instance sparsest case let built setting first equation entry structure matrix form next equations shift pair one position right previous one complete last equation whose form form structure built proofs hypothesis encoding unique pairs spread structure deciding whether minimal structure size corresponds exactly bpbp special case bbp minimal structure densest possible corresponding bipartite graph biclique deg vertices instance see minimal structure found second recursive step coa fig proof proposition let structure two total causal mappings proof proof based argument nayak present arguably much clearer intuitively shows differ variable equation mapped variables must causally dependent strongly coupled show reduces show first containment second understood following symmetry closure operators extensive idempotent shall idempotence suffices show must show well observe def bijections invertible functions thus trivially else disagree equations map onto show next case shall take equations let number disagreed equations let mapped variable construct sequence length element defined defining sequence equation disagreed mappings immediately followed ars symmetrically proofs hypothesis encoding sequence form since must codomain must repetition point sequence index obviously else note must also codomain codomain let point sequence easy see either thus transitivity causal chain must eventually finally since ars transitivity proof theorem let set defined complete structure may may canonical proof show properties def hold produced alg property may hold initialization algorithm sets inserts scanned termination obviously property holds also note ars def bijection property hold must redundant found closure lemma case bijection follows thus must case property holds finally property hold closure may find pick structure whose matrix rows instance alg encodes proofs causal reasoning let note written observe derived sufficient show property may hold extraneous removed lhs without loss information lemma let def set attributes proof lemma know iff need prove equivalent show alg xclosure gives correct answers known theorem ullman note xclosure inserts lemma let set iff attribute closure proof ullman let suppose definition must follows union well conversely suppose decomposition proofs causal reasoning proof lemma let complete structure total causal mapping set encoded given attribute folding exists proof existance ensured degenerate case empty application fact folded folding exists else folded yet proofs causal reasoning theorem def must def finite application derive although may many intermediate attribute sets along transitive chaining satisfying conditions claim least one folding suppose leading infinite regress nonetheless far cycles ruled force def must infinite number finite bounded therefore folding must exist moreover observe encoded def bijection exactly one attribute set straightfoward rationale led infer folding existance note must single chaining cycles ruled force def finite folding unique proof theorem let complete structure set encoded given let afolding correctly computes attribute folding time proof proof roadmap note afolding monotone size increase terminates precisely denotes attributes step outer loop folding step shall prove induction given attribute returned afolding unique attribute folding base case theorem exactly one attribute set algorithm always reaches step base case placed therefore fact empty application specifically must folded set consumed step fact def must folded proofs causal reasoning step induction let assume lemma know unique folding step inductive step suppose placed since yet consumed write assumed application consuming claim folded suppose def must since must note decomposition well lemma case means already consumed though finally time bound note worst case exactly one consumed step outer loop let decreased stepwise arithmetic progression scans required overall note also however may symbol read scan worst case symbols read thus measure actually overestimated therefore alg bounded proof corollary let complete structure set encoded given algorithm folding correctly computes folding time time complexity alg proof theorem know alg afolding correct terminates alg folding necessarily inserts initialized empty exactly one scanned thus termination afolding correct know unique folding proofs causal reasoning therefore must case alg correct finally time bound algorithm iterates without read symbols step afolding takes time thus folding takes know theorem remark takes proof proposition let complete structure total causal mapping set encoded given let folding proof lemma know attribute folding exists unique thus folding automatically satisfies def long show canonical def moreover theorem know singletonrhs consider lemma afolding builds bijection mapping exactly one since obvious well covers attributes rhs also bijection implies since exactly one attribute rhs lemma finally show unlike folding must suppose since rhs must suppose folded note also zimplies transitivity note also implies assumed folded must folded folded though attribute folding unique even though know lemma must unique thus must altogether therefore parsimonious proofs causal reasoning proof theorem let complete structure total causal mapping set encoded given causally dependent iff proof prove statement induction consider first direction converse base case let theorem default empty application alg ensures exactly one equation ars force must thus obviously well induction recall armstrong rule adapted particular case sets inductive hypothesis take two hcz assume causal dependency property holds attributes encode variables let note moreover assumed derived also satisfy condition theorem easy see property holds likewise nontrivial fact implies also inductive hypothesis either must converse direction shown symmetrical inductive argument base case suppose know ars moreover case alg ensures must thus empty application proofs causal reasoning inductive step shows property still holds arbitrary causal dependencies proof proposition let structure variable first cause exogenous accordingly variable first cause proof proof straightforward definitons first statement suppose contradiction exogenous first cause def bijective moreover exogenous def must endogenous words must ars hence however first cause def symmetrical argument proves second statement also contradiction take variable endogenous suppose first cause variable endogenous def must exogenous words must ars thus total causal mapping must therefore possible derive first cause assumption must proof lemma let complete structure total causal mapping set encoded given variable first cause variable either proof prove statement construction theorem def one conditions first cause moreover theorem know hold def bijective moreover since proofs causal reasoning parsimonious rhs let know hence required theorem exist derived finite application must applied hcw get hzw easy see exists second condition lemma obviously exist satisfy requirement imposed theorem proof theorem let complete structure total causal mapping set encoded given let attribute encodes variable every first cause encoded attribute proof show existance missing first cause folded leads contradiction suppose contradiction missing first cause lemma since variable first cause variable must exogenous either first case since exogenous parsimonious def consumed implies must however assumption def yet second case observe let get hzs well either folded rendering two cases analysis folded folded note folding taken endogenous variable proofs probabilistic synthesis parsimonious lemma folding must unique therefore must else assume folded def however decomposition must either nontrivial know latter case argument used first case exogenous parsimonious must furthermore get easy see situation recurs folded eventually folded like lemma uniqueness folding proofs probabilistic synthesis proof theorem let resp complete structure big fact table hypothesis let repaired factorization explanation table hypothesis recorded let schema defined bcnf proof let yki ykj resp projection predictive projection note either form form must show violate yki ykj easy see projection def onto ykj empty like projection onto yki projections note def violate bcnf yki must superkey yki note proofs probabilistic synthesis superkeys yki also definition problem know maximal group must superkey yki thus projection subject bcnf violation predictive projection ykj let reconstruct process towards deriving note derived simulating applications note also cyclic involved applications must always result alg merge projection thus nontrivial projection addition must form rendered decomposition superkey ykj therefore predictive projection subject bcnf violation minimality note consequence alg merge two schemes ykp ykq rendered iff case hold prove find merging pair arbitrary schemes shall hinder bcnf fact take ykp ykq xzv neither superkey therefore bcnf proof theorem let complete structure hypothesis big fact table repaired factorization explanation table hypothesis recorded let schema defined join subset projections lossless predictive projection ykj result join theoretical big fact table turn proofs probabilistic synthesis projections yki lossless proof item lemma know pair yki ykj projections lossless join iff hold def know fact repaired therefore yki ykj lossless join since join associative operation chosen yki ykj arbitrarily clearly subset projections must lossless join item predictive projection ykj take join yki projections pivot attribute representing first cause item know join lossless must show join yki big fact table also lossless lemma case iff hold fact trivially finally join theoretical big fact table must lossless likewise fact note also trivially well since join commutative order application irrelevant therefore join joins examined taken together must lossless lemma let set attributes relation schemes let projection onto lossless join iff hold proof see ullman
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aug limitations transversal computation quantum homomorphic encryption michael yaoyun department mathematics department electrical engineering computer science university michigan ann arbor usa mgnewman shiyy abstract transversality simple effective method implementing quantum computation faulttolerantly however quantum code qecc transversally implement quantum universal gate set eastin knill phys rev since reversible classical computation often dominating part useful quantum computation whether implemented transversally important open problem show small set codes rule binary qecc transversally implement classical reversible universal gate set particular qecc implement toffoli gate transversally prove result constructing information theoretically secure inefficient quantum homomorphic encryption scheme inspired ouyang homomorphic encryption allows implementation certain functions directly encrypted data homomorphically scheme builds almost qecc implements code transversal gate set homomorphically observe restriction imposed nayak bound focs implying quantum fully homomorphic scheme implementing full set classical reversible functions must highly inefficient scheme incurs exponential overhead qecc implementing toffoli transversally would still violate lower bound scheme introduction restrictions transversal gates transversal gates surprisingly ubiquitous objects finding applications quantum cryptography quantum complexity theory course quantum instability quantum information quantum codes allow encoding single qubits multiple qubit systems errors small subsets physical qubits corrected performing computations codes carries risk propagating errors different subsystems unless code implement computation way preserves subsystem structure informally types logical operators decompose product across subsystems called transversal theorem limits ability quantum codes prevent error propagation theorem quantum code implement quantum universal transversal gate set transversal gate sets valuable models quantum computation implement associated transversal gate sets free incurring comparatively significant overhead often form magic state distillation gauge fixing recently deconstructions gates pieces one implement remaining gate set making computation space universal improving efficiency overhead designing new fault tolerant architectures supplement transversal gates central quantum fault tolerance implementing classical reversible computation efficiently would extremely desirable many quantum algorithms primarily classical subroutines relatively small number quantum gates several proposals example factoring cryptographically large rsa key using shor algorithm requires around toffoli gates perform modular exponentiation alone dominating portion circuit toffoli universal classical reversible computation one might ask quantum codes naturally implement toffoli thus classical computations transversally give restrictions ability qeccs theorem informal almost quantum code implement classical universal transversal gate set particular almost quantum code implement toffoli gate transversally exceptions theorem distance codes decompose product states logical computational basis subcode fails erasurecorrecting essentially one think maximally redundant quantum codes concatenation repetition code distance inner code similar shor stabilizer code written product ghz states expect code implement toffoli transversally remains case proof technique rule particular proof apply binary additive codes result perhaps slightly surprising since exist qeccs triorthogonal codes implement ccz gate transversally fact transversal toffoli gates map different quantum codes increasing degree underlying polynomial quantum homomorphic encryption main ingredient proof secure homomorphic encryption scheme generally homomorphic encryption means delegating computation sensitive data securely allows encryption data way another party perform meaningful computation ciphertext without decoding preserving security underlying plaintext scheme termed fully homomorphic encryption fhe implement universal class functions computation space recently extensions homomorphic encryption quantum setting considered instead encrypting classical data implementing addition multiplication gates homomorphically quantum homomorphic encryption aims encrypt quantum data implement unitary gates homomorphically progress made recently extended work leveled scheme could homomorphically implement quantum circuits aforementioned schemes computationally secure since use classical fhe subroutine great indictment fhe built difficulty certain hard lattice problems leading candidates encryption however quantum information often promises security guarantees impossible classically intermediate advances also made restrictive setting one scheme allows implementation large class unitaries homomorphically less stringent guarantees recently proposed compact scheme size encoding scales polynomially size input limited clifford circuit class scheme achieves strongest notion imperfect probability distinguishing two ciphertexts exponentially suppressed size encoding scheme based noisy quantum encoding data take encoding circuit particular quantum code replace ancilla bits encoding uniformly random noise encryption choosing random embedding code yet uniformly random noise scheme links transversal gates transversal gates code exactly gates implemented homomorphically limitations fundamental limitations homomorphic encryption known purely classical scheme efficient impossible violating lower bounds setting single server private information retrieval shown best case scenario mutual information plaintext ciphertext precisely zero efficient quantum fhe impossible result actually applies restrictive setting classical data encrypted quantum data allowing classical reversible functions evaluated homomorphically ask whether relaxing imperfect might allow efficient unfortunately case proposition informal efficient impossible concurrent work proposition observed provide precise statement proof restriction appendix result seen combining proof technique similar single server private information retrieval bounds quantum setting essence inefficiency follows viewing classical data using quantum encoding certain quantum random access encoding qrac see function class wish implement homomorphically bounds qracs place lower bounds encoding size scheme precluding efficiency using variant codebased scheme proposed argue almost qecc implementing toffoli gate transversally would yield scheme violating lower bound worth noting similar tasks blind quantum computation computing encrypted data allow solutions cost interactivity client server allow interactivity definition homomorphic encryption comparison related works five works closely resemble results place restrictions transversal gate sets qeccs use similar constructions roughly summarize results compare zeng first place restrictions quantum universal transversal gate sets additive quantum codes elucidating stabilizer group structure work classified set diagonal gates implement one two qubit logical operations stabilizer codes shortly thereafter showed qecc transversal gate set must finite approximate arbitrary precision full unitary group intuitively make lie type argument showing infinitesimal transversal operations linear combinations local error operators since unitaries must act identically codespace follows group transversal operations must finite recently placed restrictions general class topologically protected logical gates topological stabilizer codes include transversal gates optimal subset showed topological stabilizer code defined lattice gate must lie dth level clifford hierarchy results extended general stabilizer subsystem codes appendix detail arguments used rule classical reversible transversal computation subclass stabilizer codes direction gave compact efficient scheme restricted class clifford circuits using magic state injection complete universal gate set adding however client server communicate protocol must limit circuits using constant number encryption noisy encoding data code followed secret embedding random noise encryption able generate indistinguishable outputs using polynomial overhead input size recent work authors independently observe proposition take positive approach arguing done spite limitation extending schematic particular codes using code concatenation achieve security polynomial overhead achieves larger circuit class iqp probably classically simulable stringent lower bounds placed nayak actually forgo noisy encoding circuit embed qeccs directly random noise removing correctable set qubits effect increasing overhead exponential factor order achieve security thanks roomy lower bound factor still small allow scheme argue directly security scheme using nonlocality quantum information encoded almost qecc idea conceptually simple order obtain encryptions data secure sufficiently short must inject randomness encodings withholding qubits code ordinarily would negatively affect correctness homomorphic evaluation property allows inject randomness still maintaining perfect recoverability intuitively spreading information across subsystems limits complexity class logical operators couple subsystems transversal operators differs fundamentally approaches quantitative bound without drawbacks however maximally redundant codes fail spread information sufficiently prototypical example shor code concatenation code however argue directly using stabilizer group structure additive code implement toffoli transversally preliminaries quantum information quickly review standard notation followed less standard tools need quantum information theory complete view see throughout working qubit quantum systems denote log dim number qubits constituting state space define general quantum state positive operator trace one call state pure rank otherwise call mixed note operator mixed operator use notation indicate operator unclear space state lives denote state space superscript also sometimes adopt notation trb slight abuse notation also adopt convention permutation also indicate unitary permutation operator corresponding physical permutation qubits also sometimes omit dimension identity operator usually dimension implicitly trace normalization factor acts space dimension norm refers usual schatten singular values kakp recall api max think means bounding ability distinguish two quantum states refers positive semidefinite partial ordering collection quantum states indexed sometimes write denote expectation uniformly random choice regularly referring several particular gates list toff ccz definition random access code qrac mapping string quantum state along family measurements satisfying mij generally consider protocol retrieving call protocol ith query qrac satisfying definition quantum code simply subspace hilbert space along fiducial orthonormal logical basis let denote projection onto code called quantum code qecc subspace dimension satisfying operators hermitian call correctable errors say distance code operationally means exists recovery channel error acting nontrivially qubits corrected recall also qecc correct errors known locations call types errors erasure errors call codes satisfying codes simplicity restrict discussion qeccs encoding single qubit qeccs quick review proof shows make assumption without loss generality arguing classical universal transversal gate set single encoded logical qubit similar reasoning proof applies subsystem codes well require logical operators code independent state gauge qubits collection logical qubits encoded code decompose physical qubits fixed partition every partitioning set contains exactly one qubit code block refer partitioning sets subsystems collective code definition stabilizer group abelian subgroup pauli group containing additive stabilizer code encoding logical qubits described simultaneous pauli operators comprising stabilizer group generators logical pauli operators code correspond normalizer cosets follows distance code minimal weight operator definition quantum error correcting code define logical states physical encoding define logical gate codespace preserving physical gate satisfies set transversal gates associated logical gates decompose product across subsystems say length code acts single subsystem codespace preserving map code gate define logical gate strongly transversal decomposes following example allow coordinate permutations definition transversality definition say quantum code spanc code written vector decompose product state across bipartition additionally assume occupy subsystem makes sense refer span jth subcode assumptions natural justify discussion code additionally qecc subcode distance simply call resulting code maximally redundant code note pure state code least code guiding example shor code seen concatenation repetition outer code complementary ghz inner code neither quantum erasure correcting case subcodes identical maximally redundant code concatenation repetition code distance subcode intuitively codes erase enough qubits mix state still remaining perfectly correctable redundancy used classical codes protect information quantum codes must spread information protect sense codes maximally redundant spread information least show maximally redundant codes maximally redundant codes subcodes comprised subspaces binary qeccs hope implementing logical toffoli transversally homomorphic encryption define scheme three algorithms performed two parties call client server restrict limited setting quantum scheme implementing boolean functions classical data using quantum encodings course impossibility result extends difficult task quantum computations quantum inputs parameters scheme given gate set formally define algorithms scheme acting client private workspace message space sent client server encryption message space sent server client evaluation client chooses input encrypts private randomness obtain assume quantum evaluation key appended encryption client sends message portion encryption server define length message size encoding server description circuit applies evaluation map encrypted state possibly consuming evaluation key process server sends portion state back client define length message size evaluated encoding iii client decrypts returned evaluated encoding using side information recovers associated plaintext encryption scheme certainly satisfy scheme three additional properties scheme satisfy well security inputs letting denote outputs thinking states mixtures encryptions uniformly random choices secret key circuit input probability randomness protocol subscript denotes first bit output restriction first bit argue directly boolean functions ease exposition also assume without loss generality protocols perfectly correct allow call scheme fully homomorphic homomorphic set classical boolean circuits iii compactness priori server could nothing except append description circuit run decryption function decrypting avoid trivial solutions like demand total client actions protocol scale complexity functions evaluated fixed function size input intuitively captures motivation behind homomorphic encryption limiting computational cost client however note standard definition compactness refers decrypt function specifically denote scheme homomorphic class functions satisfying properties scheme set boolean circuits denote scheme scheme observe scheme must inefficient precise statement proof see appendix proposition communication cost must exponential size input coding based scheme consider strategy implementing compact qhe using quantum codes simple block embedding encryption scheme homomorphically implementing quantum circuits classical input similar construction use property withhold correctable set qubits encoding coding qhe scheme arguments qecc size noise code block secret key input encode pure state logical computational basis defining let collection subsystems comprised one subsystem subcode form trr essentially state collection codewords codeword missing one subsystem subcodes initialize arrays maximally mixed qubits replace column array subsystem forms encrypted state publish constant number labeled encryptions used ancilla homomorphic evaluation figure description encryption procedure code based qhe scheme encryption bbbb encoding figure diagram illustrating qhe scheme quantum code withholding single subsystem subsystem remains hands client arrows connecting subsystems indicate subsystem column mapped filled dots represent code qubits empty dots represent maximally mixed qubits scheme detailed figures using notation summarize encryption channel defined secret key input string sometimes use notation instead instead omitting unconcerned underlying plaintext total size encrypted input mnp qubits preceding notation described scheme parameters mnp mnp implementing set gates homomorphically lemma let encryption scheme detailed figure let denote group transversal operators associated underlying quantum code proof let logical operator wish apply codestate definition implies decomposed product operator operator acts subsystem code without knowledge secret key third party implement applying operator operator local subsystem server possession say one columns corresponding array returning resulting data party secret key party decrypt obtain state form supported subsystems client withheld since viewing erasure error subsystems exists recovery channel decoding obtain desired note scheme compact qhe scheme since recovery decryption channel depend complexity aim compute security proposed scheme namely tradeoff size input size encoding mnp guarantee avoid confusion point code size constant concatenating achieve security amplifying size noise embedding want show scheme inefficient parameters still defeat nayak bound simplify security proof use stringent requirement outputs indistinguishable uniformly random noise see nonlocality information stored qeccs essential allowing withhold qubits still delegating computation server imposes requirement using quantum codes evidenced following observation lemma suppose replace preceding scheme one withhold physical qubits comprising pure state code must bounded away zero proof counting rank encrypted state note rank rank thus fraction nonzero eigenvalues must log since log fraction nonzero eigenvalues goes zero kes must bounded away zero claimed security tradeoff qhe coding scheme aim give inefficient sufficient security parameters coding qhe scheme argue qecc implementing sufficiently large transversal gate set set classical reversible gates would violate nayak bound parameters first need small lemma structure partial trace operator proof found appendix lemma hilbert space decomposition ready prove security tradeoff adopt notation used proposed scheme convenience note demanding stronger condition outputs indistinguishable random noise proposition scheme described figure letting dimension subsystem proof third line follows noting quantum state write denote size intersection considered sets decompose prs note furthermore permutation coordinates may write dim noting quantum state trace one multiplicativity trace tensor products next consider general case permutation coordinates subsystem intersection final line follows lemma withheld subsystem subcode underlying qecc row mixed follows separability across encoded qubit multiplicativity trace across tensor products follows exists putting together observe including first term sum get desired limitations classical transversal computation left two competing bounds one hand follows nayak bound appendix encryption scheme security communication size log choose parameters leak constant fraction information input see chosen fixed function input size must log using notation parameters aforementioned coding scheme means mnp log restriction functions inputs note assume ancilla overhead since constant gets absorbed asymptotic bound construction scheme transversal gate set underlying choice quantum code next would like choose function suffices choose function lim equivalently require select still plugging back nayak bound see asymptotically log log size function class seen function returning number unique members class inputs particular set boolean functions log shows code satisfying hypotheses scheme implement toffoli transversally justify earlier assumptions structure candidate codes suppose qecc could implement logical toffoli gate transversally first note tensor decomposition logical states must align else restriction logical toffoli one element product would unitarily map pure state mixed state furthermore think qecc criterion definition diagonal condition since paulis form operator basis always assume element pauli group codes logical basis states becomes note trace corresponding subsystem obtain code correctable error set complement system furthermore trace subcode subsystems obtain code correctable error set complement observations follow noticing subcodes must satisfy diagonal condition follows security proof code would satisfy hypotheses scheme violate lower bound proposition thus furthermore logical transversal toffoli entire code must restrict global phase logical transversal toffoli gate subcodes definition thus subcode must distance summarize theorem qecc maximally redundant code admit classicalreversible universal transversal gate set particular code implement toffoli gate transversally note also scheme figure negligible summarizing parameters coding scheme proposition quantum code transversal gate set described protocol compact quantum encryption scheme security negl input size encoding size highly inefficient pause give intuition suits purposes one hand envision trivial hiding schemes encoding length bit nayak bound allows higher efficiency roughly demanding encodings implementing set classical functions bits homomorphically must length least bit finally scheme encoding length efficient enough defeat bound allow argue theorem maximally redundant codes simple design assume additive use additional stabilizer structure argue directly implement logical toffoli transversally observation directly obtain following corollary additive qecc implement transversal toffoli proof see appendix note also follows arguments appendix central result follows corollary qecc maximally redundant code implement toffoli gate transversally finally note concatenating code code remains distance must increase least furthermore code implements toffoli strongly transversally concatenation result observe following corollary qecc implement strongly transversal toffoli discussion exist maximally redundant codes implement toffoli transversally one essentially think qeccs formed concatenating outer repetition code distance inner code stabilizer subspace intuitively since inner code quantum code spreads information one basis precisely inner code satisfies diagonal qecc criterion less restrictive condition still must complementary outer code allows argue impossibility additive case unfortunately comparison structure general codes less particular know examples code expect qecc implement toffoli transversally view exception consequence lack structure general codes hope resolve exception upcoming work qhe scheme detailed compact highly inefficient immediate question would refine security proof uses strong security demand would interesting see modified approach achieve efficient transversal gate sets general quantum codes size encoding fixed polynomial input length certain quantitative properties nonlocality qeccs see might helpful endeavor following outline could also expect extend scheme built code desirable transversal gates accommodate constant number gates one might tailor qecc specific algorithm makes heavy use transversal gate set one might also tailor scheme homomorphically implement algorithm furthermore would theoretical interest find protocol matching lower bound implicit proposition another interesting open question consider leveled schemes allow client preprocessing scale size circuit relaxation allow efficient universal schemes polynomial sized circuits mirroring computational security case first step might try apply techniques instantaneous nonlocal computation proved invaluable computationally secure scheme moreover ways converting codes together form universal transversal gate set clear implement strategy since noisy embedding present barriers measuring syndromes elements taken together might useful extending current scheme finally one could ask correspondence transversal gates quantum codes nontrivial gate sets based richness function classes realize particular asked maximum size finite group implemented logically transversally indeed since clifford group size one could reasonably expect efficiently implement clifford gates homomorphically information theoretic security done hope arguments might extend past classical reversible circuit classes address question although unclear generalize nayak bound apply general finite subgroups unitary group acknowledgments authors would like thank cupjin huang fang zhang useful discussions particular concerning security proof grateful audra mcmillan anne broadbent zhengfeng comments earlier draft paper particular thank anne bringing attention pointing proposition follows result paper argument similar classical case also thank fernando pastawksi pointing lemma research supported part nsf awards references ambainis leung mancinska ozols quantum random access codes shared randomness october anderson connor classification transversal gates qubit stabilizer codes quantum information computation arnaud cerf exploring pure quantum states maximally mixed reductions january phys rev baumeler broadbent quantum private information retrieval linear communication complexity april journal cryptology volume issue bombin gauge color codes optimal transversal gates gauge fixing topological stabilizer codes august new phys bravyi koenig classification topologically protected gates local stabilizer codes phys rev lett bremner jozsa shepherd classical simulation commuting quantum computations implies collapse polynomial hierarchy august proceedings royal society volume issue broadbent delegating private quantum computations june canadian journal physics broadbent fitzsimons kashefi universal blind quantum computation proceedings annual ieee symposium foundations computer science focs broadbent jeffery quantum homomorphic encryption circuits low complexity proceedings advances cryptology crypto broadbent song watrous proof systems qma april proceedings ieee annual symposium foundations computer science focs cramer ducas peikert regev recovering short generators principal ideals cyclotomic rings cryptology eprint archive report cross quantum computer architectures using hierarchies quantum codes phd thesis dulek schaffner speelman quantum homomorphic encryption polynomialsized circuits august crypto advances cryptology crypto eastin knill restrictions transversal encoded quantum gate sets july phys rev lett fillinger lattice based cryptography fully homomorphic encryption http fisher broadbent shalm yan lavoie prevedel jennewein resch quantum computing encrypted data january nature communications article number fowler devitt jones surface code implementation block code state distillation january scientific reports fowler mariantoni martinis cleland surface codes towards practical quantum computation august phys rev gentry fully homomorphic encryption scheme thesis stanford university gottesman stabilizer codes quantum error correction caltech thesis jones composite toffoli gate error detection march phys rev jones novel constructions toffoli gate phys rev knill laflamme theory quantum codes april lai chung quantum homomorphic encryption preparation nautrup friis briegel topological code switching two dimensions september nayak optimal lower bounds quantum automata random access codes april focs nielsen chuang quantum computation quantum information cambridge university press new york ouyang tan fitzsimons quantum homomorphic encryption quantum codes august ozols notes clifford group july http paper paetznick reichardt universal quantum computation transversal gates error correction april phys rev lett pastawksi yoshida logical gates quantum codes phys rev peikert decade lattice cryptography march foundations trends theoretical computer science scott multipartite entanglement codes entangling power quantum evolutions may phys rev speelman instantaneous computation low quantum circuits november tan kettlewell ouyang chen fitzsimons quantum approach homomorphic encryption sci yoder takagi chuang universal gates concatenated stabilizer codes march phys rev fitzsimons limitations information theoretically secure quantum homomorphic encryption june phys rev zeng cross chuang transversality versus universality additive quantum codes september ieee transactions information theory volume issue result set show efficient impossible define state icm note ihsx always purify system size without loss generality may assume size message sent client server next information theoretic security state encryption subsystem must almost independent formally ktrc ihsx trc security scheme equivalently exists vxc defining vxc vxc ihsx furthermore homomorphic property abbreviating fev fev replacing definition contractivity trace defining distance also elucidate underlying qrac define mapping let denote query note vxc vxc index thinking length bit string xth bit defined set boolean functions communication cost protocol recall bound efficiency qracs theorem nayak bound exists binary entropy function must total communication cost protocol security allowing noting see communication cost thus either size encoding evaluated ciphertext must exponentially long input precluding efficiency short proposition communication cost must exponential size input proof lemma proof expanding terms outer products xxx xxx hand xxx claimed proof corollary proof theorem suffices consider maximally redundant codes suppose sake contradiction additive code could implement toffoli transversally let denote group commutator denote states operations acting subcodes full code assume subcode speak directly inner outer codes general argument follows similarly since code additive code distance minimal weight logical pauli operator acting code multiplicativity inner product tensor products since outer code distance follows qecc criterion must weight least must weight since underlying inner code distance assumption outer classical repetition code factors tensor product transversal toff outer code must restrict global phase transversal toffl inner code since working multiqubit gates let denote logical gate acting ith code block compute directly toffl furthermore toffl transversal follows representative also transversal supported subsystems support similar argument must also contained subsystems supporting turn already observed minimal weight representative must least contradiction representative weight alternate proof stabilizer codes offer alternate proof limiting universal transversal reversible computation subclass stabilizer codes arguments based results reproduce completeness definition clifford hierarchy sequence gate sets defined recursively define pauli group note clifford group fails group note reversible circuits saturate clifford hierarchy fact lie outside entirely gate gate lies toffoli simply lies third level clifford hierarchy next recall stabilizer cleaning lemma found lemma let stabilizer code let subset physical qubits code logical operator supported acts trivially logical operator exists representative supported call subsets cleanable equipped cleaning lemma summarize following lemma lemma let stabilizer code let set cleanable subsets physical qubits comprising let logical operator supported transversal respect proof proceed induction base case logical operator supported cleanable subsets let logical pauli operator cleaned let denote group commutator since stabilizer code logical pauli operators transversal supp cleanability implies cil since true must similarly suppose supported cleaning logical pauli see supp inductive hypothesis implies logical pauli thus completing proof argument generalizes subsystem codes refer reader complete description consequence obtain following corollary stabilizer code implement classical reversible universal transversal gate set proof partition code block single subsystem subsets length code since code logical operator supported single subsystem must act trivially codespace subsets cleanable lemma transversal logical gate must lie since reversible circuits saturate logically transversally implementable
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optimal densification fast accurate minwise hashing anshumali shrivastava rice university houston usa mar abstract minwise hashing fundamental one successful hashing algorithm literature recent advances based idea densification shrivastava shown possible compute minwise hashes vector nonzeros mere computations significant improvement classical advances led algorithmic improvement query complexity traditional indexing algorithms based minwise hashing unfortunately variance current densification techniques unnecessarily high leads significantly poor accuracy compared vanilla minwise hashing especially data sparse paper provide novel densification scheme relies carefully tailored hashes show proposed scheme without losing runtime efficiency significantly accurate existing densification techniques result obtain significantly efficient hashing scheme variance collision probability minwise hashing experimental evaluations real sparse highdimensional datasets validate claims believe given significant advantages method replace minwise hashing implementations practice anshumali rice edu speech text quite popular broder fetterly enriching features information leads blow dimensionality common text representations vocabulary size representation requires dimensionality representing genome sequences features consisting characters higher ondov leads around dimensions deal overwhelming dimensionality increased emphasis use hashing algorithms minwise hashing minwise hashing provides convenient way obtain compact representation data without worrying actual dimensionality compact representations directly used large scale data processing systems variety tasks minwise hashing defined binary vectors binary vectors also equivalently viewed sets universe features containing attributes corresponding entries minwise hashing belongs locality sensitive hashing lsh family broder charikar method applies random permutation random hash function given set stores minimum value permutation mapping formally min given sets shown elementary probability arguments introduction motivation recent years witnessed dramatic increase dimensionality modern datasets weinberger show dataset trillion unique features many studies shown accuracy models keeps climbing slowly exponential increase dimensionality large dictionary based representation images quantity well known jaccard similarity resemblance popular similarity measure information retrieval applications broder probability collision equality hash values minwise hashing equal similarity interest optimal densification fast accurate minwise hashing particular property also known lsh property indyk motwani charikar makes minwise hash functions suitable creating hash buckets leads sublinear algorithms similarity search lsh property minwise hashing popular indexing technique variety data processing applications include duplicate detection broder henzinger similarity bayardo temporal correlation chien immorlica graph algorithms buehrer chellapilla chierichetti najork recently shown lsh property minwise hashes used generate kernel features learning minwise hashing known theoretical optimal many scenarios bavarian furthermore recently shown provably superior lsh angular similarity cosine similarity compared widely popular signed random projections shrivastava unique advantages make minwise hashing arguably strongest hashing algorithm theory practice hashing cost bottleneck first step algorithms relying minwise hashing generate large enough minwise hashes fingerprints data vectors particular every data vector repeatedly computed independent permutations hash functions hashes used variety data mining tasks cheap similarity estimation indexing search kernel features large scale learning etc computing hashes vector traditional minwise hashing requires computation number vector computation multiple hashes requires multiple passes data number required hashes typically ranges hundreds several thousand example number hashes required famous lsh algorithm grows size data showed necessity around hashes per data vector learning hashing time main computational resource bottleneck step almost applications using minwise hashing related fast sketches lsh two notable techniques estimating jaccard similarity sketches one permutation hashing although two sketches cheap compute satisfy key lsh property therefore unsuitable replacing minwise hashing shrivastava also substantial empirical evidence using sketches indexing leads drastic bias expected behavior leading poor accuracy idea densified one permutation hashing cently shrivastava showed technique densifying sparse sketches one permutation hashing provably removes bias associated one permutation hashing please see section details first success creating efficient hashing scheme satisfies lsh property analogous minwise hashing time complete process requires computations instead traditional bottleneck efficient scheme directly translates algorithmic improvements variety machine learning data mining tasks current densification inaccurate sparse datasets densification process although efficient unbiased shown unnecessarily higher variance shown shrivastava traditional densification lacks sufficient randomness revealed densification could provably improved using extra random bits improved scheme reduced variance retained computational efficiency see shrivastava details improved variance associated significant performance gain task search work show even improved densification scheme far optimal findings shrivastava leaves open curiosity best variance achieved densification without sacrificing running time close providing scheme contributions show existing densification schemes fast minwise hashing suboptimal worse variances zero increasing number hashes variance increase number hashes converges positive constant behavior implies increasing number hashes point lead improvement popular belief accuracy randomized algorithms keeps improving increase number hashes circumvent issues present novel densification scheme provably superior variance compared existing schemes show proposal optimal variance achieved densification furthermore variance new methodology converges zero increase number hashes desirable behavior absent prior works proposal makes novel use hashing could independent interest benefits improved accuracy come loss computational requirements scheme retains running time efficiency densification provide rigorous experimental evaluations existing solutions concerning accuracy running time efficiency real datasets experiments validate theoretical claims show significant optimal densification fast accurate minwise hashing provement accuracy comparable minwise hashing significant gain computational efficiency important notations concepts equation lsh property leads estimator jacard similarity using hashes defined indicator function paper variance mean variance estimator notations like mean variance estimator used hash function denote set integers denotes number points samples dataset used dimensionality use min denote minimum element set permutation applied set another set hashing generate hashes generally different bins since distribution properties drop subscripts use denote hashing schemes shrivastava shrivastava respectively background fast minwise hashing via densification hashing definitions randomized function huniv following property huniv huniv carter wegman showed simplest way create hashing scheme pick prime number sample two random numbers compute huniv mod mod one permutation hashing empty bins shown dahlgaard instead computing global minimum equation min efficient way generate sketches using one permutation first bin range space disjoint equal partitions followed computing minimum bin partition let denote ith partition range space formally ith one permutation hashes oph set defined min hop otherwise obvious computational advantage scheme likely generate many hash values requires one permutation pass sets binary vectors shown two sets conditional collision probability similar minwise hashing let hop hop however hop hop hop hop indicator random variable event ith partition corresponding empty see figure bin constant chance empty thus positive probability event given pair hence large datasets big constant fraction data consist simultaneously empty bins trials bad event happen fraction increases significantly sparsity data sparsity increases probability bad event see table statistics empty bins real scenarios unfortunately whenever outcome random permutation leads simultaneous empty bins event lsh property valid fact sufficient information present simultaneous empty partitions meaningful statistics hence one permutation hashing used lsh simple heuristics handling empty bins suggested shrivastava leads significant bias shown theoretically empirically bias leads significant deviation expected behavior one permutation hashing compared minwise hashing thus one permutation hashing although computationally lucrative suitable replacement minwise hashing idea densification shrivastava authors proposed densification reassignment values empty bins reusing information bins fix bias one permutation hashing overall procedure quite simple empty bin borrows values optimal densification fast accurate minwise hashing leads partition range bins one permutation hashing get min bin empty bin densification reassign right circular improved densification reassign left right circular depending random bits randbits figure illustration one permutaion hashing oph two existing densification schemes shrivastava densification simply borrows value nearby empty bins different sets different pattern bins since pattern random process similar random unbiased values satisfies lsh property bin shrivastava est bins towards circular right left see figure illustration since positions empty bins random shown densification reassignment equivalent stochastic reselection one hash set existing informative coming bins hashes lsh property kind reassignment restores lsh property collision probability two hashes reassignment exactly minwise hashing densification generates hashes required lsh property requires two passes one permutation sketches making total cost one permutation shrivastava also needed offset value hash bin always reset need offset use actual values hashing plus densification significant improvement classical minwise hashing led algorithmic improvement randomized algorithms relying minwise hashing hash computation cost bottleneck step lack randomness densification pointed shrivastava densification scheme shrivastava unnecessarily high variance particular probability two empty bins borrowing information nonempty bin significantly higher probability due poor randomization load balancing hurts variance shrivastava showed infusing randomness reassignment process utilizing extra random bits provably improves variance see figure example illustration method running time improved scheme computing hashes improvement retains required lsh property however time improved variance improved variance led significant savings task search real sparse datasets issues current densification careful analysis reveals variance even improved scheme still significantly higher worse even extreme case take variance converges positive constant rather zero implies even infinite samples variance zero positive limit increases sparsity dataset particular following theorem limiting variances existing techniques theorem give two finite sets limiting variance estimators densification improved densification given lim lim convergence variance constant value despite infinite samples existing densification also evident experimental findings see figure observe mse mean square error curves flat increasing similar phenomenon also reported shrivastava noted classical minwise hashing variance pair thus current densification although fast loses significantly terms accuracy remove issue densification particular show optimal densification fast accurate minwise hashing example poor load balancing densification informative value used improved densification informative values used randbits figure illustration poor load balancing existing densification strategies bins local uniformly distributed thus use information far bins densification orem limiting variance proposed optimal densification goes experimental findings suggests new variance close classical minwise hashing addition new densification retains speed existing densified hashing thereby achieving best worlds optimal densification argue even improved densification enough randomness load balancing reassignment process leads reduced variance given set densification process reassigns every empty bin value one existing bins note identity empty bins different different sets ensure lsh property consistent given pair sets particular noted shrivastava given arbitrary pair whenever given bin simultaneously empty reassignment bin mimic collision probability one simultaneously bin arbitrary reassignment borrow values ensure consistency across pairs would like point reassignment idea object dataset thus ensuring consistency although current densification schemes achieve consistency selecting nearest bin shown shrivastava lack sufficient randomness intuition load balancing figure observe many contiguous bins bins densification schemes forced borrow values bin bin example even though informative bins bins information never used local bias increases probability two empty bins get tied information even many informative bins adding random bits improves extent allowing load sharing two ends instead one bins instead however load balancing far optimal locality information sharing main culprit current densification schemes note poor load balancing change expectation affects variance significantly given pairs vectors let number simultaneous bins note random variable whose value different every pair depends outcome random formally variance analysis shrivastava reveals probability two simultaneous empty bin reuses densification scheme information probability reduced utilizing extra random bits promote load balancing quite perfect load balancing perp fect load balancing simultaneous bins probability two empty bins hitting empty bins best design densification scheme achieves maintaining consistency densification time hurt running time clear scheme even exists answer question positively constructing densification method precisely requirements furthermore show achieving sufficient limiting variance zero simple hashing help break locality information reuse allow bin borrow information far bin consistently seems natural use universal hashing hope hash function section huniv whenever bin empty instead borrowing information neighbors borrow information bin huniv hash function allows consistency across two hence preserves lsh property value huniv uniformly distributed bin equally likely thus seems break locality first thought huniv also empty continue using huniv huniv reach bins whose value one issue cycles huniv chance creates cycle optimal densification fast accurate minwise hashing ment infinite loop cycle even occur huniv huniv huniv empty note process runs finds bin however cycles concern even manage get away cycles scheme provide required load balancing independent huniv attemptj due hash function huniv thus probability two empty bins reuse information bin careful inspection reveals significant chance huniv empty given set observe huniv empty bound reuse information nonempty bin empty bins huniv note control positions empty bins fact cycles happen difficult show simple assignment using universal hashing equivalent original densification order bins reshuffled using huniv worse variance denote final hashes generated proposed densification scheme algorithm using optimality formally optimal densification following fix carefully tailored hashing algorithm optimal densification input one permutation hashes hop input huniv initialize hop else attempt next huniv attempt hop next attempt next huniv attempt end hop next end end return turns way use universal hashing ensures cycles well optimal load balancing describe complete process algorithm key use hashing huniv takes two arguments current bin needs reassigned number failed attempt made far reach bin second argument ensures infinite loops changes every attempt even reach bin back cycle next time visit new set bins also even huniv attempti empty bound end bin next attempt seek bin value huniv attempti analysis optimality theorem lim nemp number simultaneous empty bins quantities given nemp nemp nemp nemp nemp nemp nemp nemp nemp nemp using formula nemp precisely compute theoretical variance interesting part formally show variance proposed scheme strictly superior compared densification scheme random bits improvements theorem finally due optimal pairwise load balancing variance best possible hope independent reassignments formally theorem among densification schemes reassignment process bin independent reassignment process bin algorithm achieves best possible variance note variance reduced allow correlations assignment process example force bin bin pick bin reassignments reduce beyond perfectly random load balancing however tied reassignment require value memory computations generating structured hash functions optimal densification fast accurate minwise hashing pair pair pair pair pair pair pair pair sim table statistics pairs vectors binary similarity sparsity running time show expected running time proposal including constants similar running time existing densification schemes given set interested computing hash values first step involves computing one permutation hashes sketches requires single pass elements takes max time densification algorithm requires loop size within loop bin empty requires additional loop let nemp ber empty bins therefore probability loop terminate one iteration next empty therefore expected number iteration loop run binomial random variable expectation thus expected running time algorithm given nemp nemp running time nemp nemp quantity ratio number empty bins number bins generally small rarely practice observe randomly throwing items bins expected number empty bins nemp makes number sketches usually order even good concentration size sketches rarely much larger size set even times value approximately thus quantity negligible noted implementation cost densification scheme different cost proposal evaluations aim verify theoretical claims papers empirically show proposal replace minwise hashing practical purposes establish focus experiments following objectives verify proposed scheme significantly better accuracy variance existing densification schemes validate variance formulas empirically quantify impact optimal variance practice quantification change similarity sparsity verify proposal accuracy close vanilla minwise hashing verify impact running time proposed scheme existing densification schemes proposal significantly faster vanilla minwise hashing understand running time changes change sparsity accuracy objectives selected different word pairs embedding generated corpus varying level similarity sparsity use popular termdocument vector representation word statistics word vector pairs summarized table word pairs generated hashes using three different schemes densification improved densification proposed densification algorithm using hashes estimate jaccard similarity equation plot mean square error mse varying number hashes since process randomized repeat process times every report average independent runs report integer values interval noted since three schemes lsh property bias zero hence mse theoretical variance validate variance formula also compute plot theoretical value variance equation optimal scheme also understand fast methodologies compare accuracy vanilla minwise hashing also plot theoretical variance minwise hashing results figure conclude conclusion proposed densification significantly accurate irrespective choice sparsity similarity existing densification schemes especially large note plots log scale accuracy gains drastic conclusion gains optimal densification sparse data conclusion accuracy optimal densification close accuracy costly minwise hashing optimal densification fast accurate minwise hashing pair vanilla improved proposed num hashes mse mse vanilla improved proposed vanilla improved proposed num hashes vanilla improved proposed num hashes pair vanilla improved proposed vanilla improved proposed num hashes pair num hashes pair vanilla improved proposed pair num hashes pair mse pair mse pair mse mse mse vanilla improved proposed mse num hashes num hashes figure average mse jaccard similarity estimation number hash values estimates averaged repetitions optimal densification variance close costly minwise hashing significantly superior existing densification schemes note log scale url existing paper vanilla minhash table time milliseconds requires compute hashes full data using existing densification proposed densification vanilla minwise hashing minhash slower computing hashes datasets url avg dim samples table basic statistics datasets conclusion theoretical variance proposal overlaps empirical estimates seems zero validating theorem conclusion variances mse existing densification seems converge constant zero confirming theorem speed url bins bins bins table avg number empty bins per vector rounded generated one permutation hashing sparse datasets significant hashes bins empty information used indexing kernel learning fortunately efficiently densify optimal densification generated hashes variance similar minwise hashing overall computational cost significantly less compared minwise hashing compute runtime use three publicly available text datasets url dimensionality sparsity datasets excellent representative scale size frequently encountered data processing systems google sibyl chandra statistics datasets summarized table implemented three methodologies computing hashes densification scheme proposed algorithm vanilla minwise hashing methods implemented cheap hash function replaced costly permutations clever alternatives avoid mod operations employed tricks ensured efficient possible compute wall clock time required calculate hashes three datasets time include hash computation complete data data loading time included results presented table experiments done intel processor laptop ram also get estimate importance densification also show average number empty bins generated codes available http optimal densification fast accurate minwise hashing using one permutation hashing report numbers table clearly see number empty bins significantly larger hashes unusable without densification chierichetti flavio kumar ravi lattanzi silvio mitzenmacher michael panconesi alessandro raghavan prabhakar compressing social networks kdd paris france running time numbers table conclude dahlgaard knudsen mathias tejs rotenberg eva thorup mikkel hashing statistics kpartitions foundations computer science focs ieee annual symposium ieee conclusion optimal densification fast traditional densification irrespective sparsity however optimal densification significantly accurate conclusion densification scheme significantly faster minwise hashing faster computing hashes selected datasets given simplicity hope work gets adopted references bavarian mohammad ghazi badih haramaty elad kamath pritish rivest ronald sudan madhu optimality correlated sampling corr url http bayardo roberto yiming srikant ramakrishnan scaling pairs similarity search www broder andrei resemblance containment documents compression complexity sequences positano italy broder andrei charikar moses frieze alan mitzenmacher michael independent permutations stoc dallas buehrer gregory chellapilla kumar scalable pattern mining approach web graph compression communities wsdm stanford carter lawrence wegman mark universal classes hash functions stoc fetterly dennis manasse mark najork marc wiener janet study evolution web pages www budapest hungary henzinger monika rauch finding web pages evaluation algorithms sigir indyk piotr motwani rajeev approximate nearest neighbors towards removing curse dimensionality stoc dallas ping consistent weighted sampling kdd ping shrivastava anshumali moore joshua arnd christian hashing algorithms largescale learning nips granada spain ping owen art zhang one permutation hashing nips lake tahoe najork marc gollapudi sreenivas panigrahy rina less sampling neighborhood graph makes salsa better faster wsdm barcelona spain ondov brian treangen todd melsted mallonee adam bergman nicholas koren sergey phillippy adam mash fast genome metagenome distance estimation using minhash genome biology chandra tushar eugene goldman kenneth llinares tomas lloret mcfadden jim pereira fernando redstone joshua shaked tal singer yoram sibyl system large scale machine learning technical report shrivastava anshumali ping densifying one permutation hashing via rotation fast near neighbor search icml beijing china charikar moses similarity estimation techniques rounding algorithms stoc montreal quebec canada shrivastava anshumali ping defense minhash simhash proceedings seventeenth international conference artificial intelligence statistics chien steve immorlica nicole semantic similarity search engine queries using temporal correlation www shrivastava anshumali ping improved densification one permutation hashing uai quebec optimal densification fast accurate minwise hashing weinberger kilian dasgupta anirban langford john smola alex attenberg josh feature hashing large scale multitask learning icml variance little involved collision probability following unbiased estimator variance define number simultaneously empty bins proofs theorem give two finite sets limiting variance estimators densification improved densification given lim lim nemp indicator function partition event two cases depending let mjn match mje empty match events defined mjn mje proof nemp substituting value variance formulas shrivastava taking limit get expression manipulation strictly positive note mjn mje mje mjn lsh property estimator theorem difficult show lim mjn mje mje mjn mjn min nemp number simultaneous empty bins quantities given nemp nemp nemp nemp nemp nemp nemp nemp nemp nemp proof collision probability easy using simple observation values coming different bin numbers never match across disjoint different range whenever simultaneous empty bin get reassignment value must coming bin say numbers empty thus using new events mjn interested computing mje mjn notational convenience use denote event nemp expression means nemp simplify analysis first compute conditional expectation mjn expansion linearity expectation obtain min mjn min mje optimal densification fast accurate minwise hashing mjn mjn mje mje indicator functions take values hence mjn mje independent probability two empty bins chooses bin bins lower bounded achieved optimal densification values first three terms given following expression using simple binomial enpension using fact dealing indicator random variable take values min mjn min mje let probability two simultaneously empty bins finally picks bin reassignment mie mje probability uses estimators different simultaneous bin case mie mje probability know rithm uses hashing value pairwise assignment perfectly random hashing substituting terms value rearranging terms gives required expression nemp substituting value variance formulas taking limit get theorem proof value comes analysis shrivastava theorem among densification schemes reassignment process bin independent reassignment process bin algorithm achieves best possible variance
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apr collaborative targeted inference continuously indexed nuisance parameter estimators cheng division biostatistics berkeley antoine chambaz umr paris descartes mark van der laan division biostatistics berkeley april abstract suppose wish infer value statistical parameter law sample independent observations suppose parameter smooth define two features law called comp estimate consistently fast enough product rates build confidence interval given asymptotic level based plain targeted minimum loss estimator tmle estimators would typically products machine learning algorithms focus case machine learning algorithm parameter plain tmle chosen would typically lend construction selection would empirical bias something akin empirical variance estimator opposed tmle collaborative tmle might however succeed achieving relevant prove case indeed construct show empirical processes conditions exists oracle makes bulky remainder term asymptotically gaussian asymptotically gaussian hence amenable building provided asymptotic variance estimated construction hinges guaranteeing additional well chosen estimating equation solved top estimating equation plain tmle solves optimal chosen empirical criterion guarantees wished empirical bias variance illustrate construction main result inference called average treatment effect consists marginal law conditional expectation propensity score conditional probability also conduct multifaceted simulation study investigate empirical properties collaborative tmle estimated lasso bound candidate coefficients variety scenarios shed light small moderate sample properties face baseline covariates possibly positivity violation keywords empirical process theory semiparametric models introduction wish infer value statistical parameter law sample independent observations parameter smooth function data distribution assume define two features law called qand estimate consistently fast enough joint rate build confidence interval given asymptotic level based plain targeted minimum loss estimator tmle typically parameter depends law whereas canonical gradient depends law qand estimators would typically products machine learning algorithms focus case machine learning algorithm parameter possible construct estimator lend construction targeted fashion algorithm estimation resulting estimator parameter interest literature overview general problem address often encountered observational studies effect exposure instance one wishes infer average effect exposure necessary account fact level exposure fully randomized observed population pivotal object interest studies called exposure mechanism conditional law exposure given baseline covariates example generally call law experiment wide range estimators average effect exposure require estimation propensity score estimators estimators based propensity score matching stratification estimator relying efficient influence curve among inverse probability exposure weighted estimators estimators built based targeted minimum loss estimation tmle methodology common methods estimation propensity score multivariate logistic regression propensity score adjustment variety machine learning algorithms except called collaborative variant tmle discuss shortly estimators propensity score derived preliminary step regardless essentially needed used subsequent step problematic optimality preliminary step little relation optimality subsequent step instance optimal estimator propensity score preliminary step might take values close zero therefore disqualifying viable estimator subsequent step mention optimal one less dramatic scenario using instrumental variable influences exposure outcome estimate propensity score could concomitantly yield better estimator thereof increase variance resulting estimator effect exposure prompted development called collaborative version targeted minimum loss estimation methodology estimation separated parameter main interest anymore concretely collaborative tmle ctmle consists building sequence estimators selecting one optimizing criterion targets parameter main interest instance less dramatic scenario covariates strongly predictive exposure outcome would removed resulting less bias estimator parameter main interest methodology adapted wide range fields including genomics survival analysis clinical studies derivation estimators often computationally demanding scalable versions also developed authors propose algorithm uses regression shrinkage exposure model estimation propensity score sequentially reduces parameter determines amount penalty placed size coefficient values selects appropriate parameter methodology continuously collaborative targeted learning develop article encompasses algorithm statistical analysis sheds light assumptions would provide valid statistical inference present study builds upon methodology also studied latter example application point introduction wish formalize problem stake follows recasts introductory paragraph theoretical framework adopt article setting scene let independent draws law set view element statistical model collection plausible laws know smaller primary goal infer value parameter namely statistical analysis asymptotic number observations consider case pathwise differentiable every respect tangent set exists every exists submodel satisfying iii dpt log submodel score function equals real valued mapping differentiable derivative equal shorthand notation measurable assumed moreover every associated two possibly features unrelated variation independent knowing anything tells nothing vice versa depends iii depends mapping early stage introduce pivotal every notation justified wish think expression remainder term fact depend consider case parameter stand exists universal positive constant remainder term satisfying said let algorithm estimation true law likewise let open interval closure contains algorithm estimation formally view mappings dirac respectively learn empirical measure estimators set superscript stands initial element model equal let natural estimator derived mere substitution targeted toward inference sense none known features derived specifically sake ultimately estimating toward well documented tmle literature one way target way build achieved way infer modified fluctuating procedure develop details specific example studied article estimator satisfies asymptotic expansion convention agree small values correspond less bias estimator moreover assume exists consistently estimates rate also yield may turn imply asymptotic linear expansion influence function depending particular algorithms central limit theorem guarantees asymptotically gaussian focus challenging situation necessarily anticipate analysis also relevant small moderate sample sizes order derive asymptotic linear expansion similar situation would derive asymptotic expansion unfortunately reasons believe possible without targeting presentation example deferred section cooperate sense although observe estimators two share initial estimator construction latter capitalize former contrast propose build select one ratively continuum estimators form among asymptotically gaussian conditions often encountered empirical process theory organization article section lay presentation collaborative tmle state result sections consider specific example section particularize theoretical construction analysis section describe two practical instantiations estimator developed section sections carry simulation study performances comment upon results section summarize content article proofs gathered appendix presentation result state prove general result continuously collaborative targeted minimum loss estimation version theorem assumptions clarified particular example study next sections slightly abuse notation denote instead let subset indexed real parameter ranging open subset twice differentiable surely characterize setting every consider following assumptions first one indexed exists open neighborhood set twice differentiable surely moreover surely sup denoted know build way moreover know choose deterministic met holds exists addition let given exist holds moreover exists deterministic met introduced assumptions state corresponding highlevel result entail proof relegated appendix theorem asymptotics collaborative tmle result assumptions holds commenting assumptions assumption concerns specifically depends algorithms specifically smooth around particular example studied following sections counterpart concerns algorithms example show built collaboratively way met series nested assumptions smoothness functions construction notably involve algorithms understand achieving relevant observe following oracle version lim rewritten lim view thus achieving relates finding critical points assumption formalizes convergence target limit require equal target may impossible meet see condition met instance goes zero probability difference falls class probability tending one typically holds whenever product rates convergence limits counterpart example studied following sections assume existence oracle undersmoothes enough asymptotically linear estimator note pathwise differentiable similar way say oracle definition involves happens lemma met holds difficult assess whether tall order necessarily finally states distance introduced order conditions similar nature inequality reveals met collaborative tmle continuous tuning inferring average treatment effect presentation analysis section specialize discussion inference specific statistical parameter called average treatment effect section introduces parameter recalls corresponding section section describes uncooperative construction continuum uncooperative tmles section argues selection one uncooperative tmles unlikely yield well behaved asymptotically gaussian estimator product rates convergence estimators limits fast enough sections present collaborative construction collaborative tmles select one among well behaved assumptions spelled section theorem assumptions specialized preliminary observe independent draws true law known takes values consider statistical model leaves unspecified law conditional law given might know conditional expectation given belongs set introduce parameter interest average treatment effect choose study provides wealth information paves way analysis variety parameters often encountered statistical literature generally every gives rise respectively marginal law conditional expectation given conditional probability given couple consisting average treatment effect given eqw notational conciseness let given every note conditional likelihood given given drawn bernoulli law parameter hence notation parameter viewed mapping pathwise differentiable every maximal tangent set efficient influence curve given recall definition easy check every writing instead slightly abuses notation justified integrating rhs reveals depends furthermore inequality holds uncooperative construction continuum uncooperative tmles initial estimator prerequisites let continuum candidate estimators indexed tuning parameter open interval convention agree small values correspond less bias estimator specifically denoting valid loss function estimation given log log log every defined assume empirical risk increases example could correspond fitting logistic linear regression maximizing loglikelihood constraint sum absolute values coefficients smaller equal refer algorithm lasso logistic regression algorithm uncooperative tmles let empirical law set arbitrarily denote element marginal law equals let conditional expectation given equal hence coincide one hand conditional expectation given hand evaluating yields estimator targeted toward inference sense none known features derived specifically sake ultimately estimating toward build way one way target achieved fluctuating following sense infer every introduce called clever covariate given every let characterized logit logit except conditional expectation given defined like moreover denoting equals clearly loss function given log log every induced holds property prompts say submodel fluctuates direction along submodel indexed minimizer optimal fluctuation empirical risk arg min existence assumed note twice differentiable strictly convex call tmle resulting estimator tmle readily seen equivalent since minimizes differentiable mapping holds moreover combined previous display yields targeted toward indeed furthermore view satisfies words finally tmles said uncooperative although share capitalize initial estimator two construction selecting one uncooperative tmles stage procedure crucial question select one tmle collection uncooperative tmles one lends construction given asymptotic level tmle necessarily writes well chosen could possibly deterministic fixed random element risk generated given divergence bernoulli laws parameters pinsker inequality holds therefore bounded away zero one implies deterministic exist two rates iii falls class tending one lemma guarantee met argument used repeatedly throughout article thus central limit theorem converges law centered gaussian law variance synergy convergences respective limits sufficient tmle used build cis argument falls apart worse expect case whether similar possible derive useful asymptotic linear expansion tmle depend whether derive asymptotic linear expansion derived maximizing likelihood correctly specified parametric model would asymptotically linear estimate argue little chance select remainder term asymptotically linear natural choice would use selector let recall derived explain believe solve problem let scheme instance could crossvalidation scheme random vector taking vpdifferent values probability proportion ones among coordinates empirical probability close let law training subsample empirical probability law validation subsample selector given arg min ebn unfortunately expect asymptotically linear heuristically trades bias variance estimator whereas wish clearly variance estimator trade bias variance smooth functional significantly smaller object collaborative construction finitely many collaborative tmles message sections uncooperative construction continuum standard tmles typically fail produce one asymptotically linear tmle product rates convergence estimators limits fast enough sections demonstrate collaborative construction continuum standard tmles produce one asymptotically linear tmle challenging situation appropriate assumptions recursive construction present collaborative construction finitely many tmles forthcoming theoretical presentation make fly series assumptions important ones emphasized argued selector sufficiently undersmooth make asymptotically linear term since assumed increases focus tuning parameters set assumed assumption call construction recursive unfolds follows initialization begin section every build using initial estimator estimator note placing star symbol parentheses suggests tentative estimator tmle specifically every define assuming exists assumption call set find marginal law empirical law conditional expectation given equals hence one hand conditional expectation given coincides hand assume minimized globally interior assumption call several minimizers largest choice observe every particular let assume addition differentiable open neighborhood assumption call consequently well defined see differentiable neighborhood moreover since minimizes previous mapping third equality holds light assumption call thus proven following equation solved complete initialization define note satisfy recall implied earlier recursion let arbitrarily chosen suppose already built assumptions also assumption call let present construction assumptions presentation similar initialization laid directly every build using initial estimator estimator specifically every define substituted substituted assumes existence set find marginal law empirical law conditional expectation given equals hence one hand conditional expectation given coincides hand assume minimized globally interior assumption call several minimizers largest choice moreover also assume differentiable open neighborhood assumption call consequently well defined see differentiable neighborhood since minimizes previous mapping assumption call holds complete presentation recursion define note satisfy discuss stop loop next paragraph collection tmles arguably built collaboratively derivation every heavily depends loop iterated stopping criterion met instantiations collaborative tmle laid section rely lasso logistic regression algorithm thus possible upper bound general may decide stop recursive construction whenever maximal number iterations reached successive tmles belong interval length smaller integers kmax small positive numbers hmin former chosen hmin latter possibly choice kmax would typically driven considerations computational time choice hmin would typically depend collection algorithms hmin meaning much undersmoothing certainly play using would suggest choosing characterizing given epn definition justified fact estimates asymptotic variance context prove section tmle selecting one finitely many collaborative tmles remains determine tmle select among collection collaborative tmles constructed section selection hinges principle recursive construction described section applied empirical measure subset complete data set starting defined even differs let defined like substituted assumptions recursively let defined like substituted assumptions recursive construction stopped derived defined like substituted collection collaborative tmles used define continuum collaborative tmles following straightforward way challenge associate simply let element closest preference larger two closer ones right middle formally set max min associate corresponding let scheme introduced section convention let max empty set collaborative tmle select given max arg min ebn words unique element smallest element larger minimizer collaborative tmle exists element otherwise equals contrast stark first glance main difference role play algorithms estimate played algorithms estimate closer examination reveals difference deeper replacing defined like section instead would make resulting alternative selector based good candidate inherent lack cooperation uncooperative tmles resulting estimator would even consistent fact motivates general methodology present instantiation includes twist consisting solving two critical equations see asymptotics hinges theorem study asymptotic properties collaborative tmle first specify two light requirement one hand eventually assume bounded away zero one yields choose hand note naturally gives rise substitution estimator eqw easy check see appendix following result holds lemma assume bounded away zero one falls class tending one since always estimate marginal law empirical counterpart thus define setting expression depend first components consider following assumptions first one related completes exists universal constant algorithm trained empirical measure take values moreover exists open neighborhood universal constant twice differentiable surely surely sup sup met denote first derivative converge respectively moreover holds assumption met tending one fall classes position state corollary theorem describes asymptotic properties collaborative tmle targeting average treatment effect corollary asymptotics collaborative tmle targeting average treatment effect suppose assumptions made sections constructing collaborative tmle given met addition suppose satisfied converges law central limit theorem corrolary implies centered gaussian law variance therefore provided estimate consistently conservatively build cis given asymptotic level sections investigate practical implementation collaborative tmle performances simulation study collaborative tmle continuous tuning inferring average treatment effect practical implementation section describe practical implementation two instantiations collaborative tmle algorithm presented studied section collection left embodied glmnet algorithm nature algorithm beginining unspecified obtained training procedure specifically never evaluate describe algorithm recall denotes empirical measure generic subset complete data set following algorithm implements theoretical procedure laid sections build sequence computing discretized version path lasso logistic regression regularization parameter ranging set provided options hence card set hmin min let equal build sequence hmin computing discretized version path lasso logistic regression regularization parameter ranging hmin using glmnet lambda set hmin step set based every hmin determine fluctuating section identify minimizer hmin define store every finally define long hmin set repeat steps recursively algorithm necessarily converges finite number repetitions step let number repetitions every set first component empirical law let element model equals collection collaborative tmles mapping hmin thus well defined recall definition scheme introduced section set arg min ebn hmin run steps hence collection collaborative tmles finally set max collaborative tmle select estimator procedure described section quite demanding computationally thus tempting try develop alternative algorithm would mimick simpler section emphasized see comment statement theorem one keys ensure existence knew compute derivative could specifically light easily achieved enriching fluctuation initial given would define introduce characterized logit logit defined like except conditional expectation given equals optimal fluctuation would indexed minimizer empirical risk arg min would result tmle construction would algorithm describe adapts procedure unfolds follows build sequence computing discretized version path lasso logistic regression regularization parameter ranging set provided option hence card let equal estimator choose arbitrarily arg min every define rudimentary numerical approximation derivative determine described substituted estimator main simulation study section present results simulation study behaviors performances two instantiations collaborative tmle described section section specifies synthetic distribution use section introduces competing estimators section outlines structure simulation study section gathers results written code makes extensive use package synthetic distribution synthetic distribution depends two parameters dimension baseline covariate nonnegative constant sampling unfolds sequentially along following steps sample centered gaussian law covariance matrix matrix akl identity matrix akk matrix zero matrix akl zero matrix set set sample conditionally bernoulli law paramater expit sample conditionally gaussian law conditional variance expectation define expit covariance matrix induces loose dependence structure components gathered independent groups one group consisting independent random variables groups consisting either two three mildly dependent random variables correlations equal either neither logit closed form expression independently competing estimators let independent draws recall characterization estimation consist definition given section step let algorithm fitting working model given distribution function standard normal law note working model necessarily notably absence definition whole data set emphasize recall obtained training never procedure consistent implementation original instantiation algorithm scalable instantiations compare estimators section following commonly used competitors unadjusted estimator called estimator called iptw estimator hcv estimator hcv plain tmle estimator see outline structure simulation study consider six different scenarios repeat independently times following steps collection triplets simulate data set independent observations drawn derive estimators sections well competing estimators presented section estimators two collaborative tmn les construct cis check whether contains building confidence intervals based collaborative tmles corollary asymptotic variances collaborative tmles write difficult estimate estimate empirical variance recall tmles take form construction therefore cis based collaborative anticipate asymptotic variances resulting cis wide however also anticipate omitted correction term second order relative main term put words difference small six scenarios three first scenarios investigate happens number covariates increases function sample size scenario increase scenario increase scenario log increase values pairs used scenarios presented table constants chosen sample size three scenarios table values scenarios scenario scenario scenario scenarios still set either keep fixed increase scenario keep fixed increase scenario finally scenario keep fixed challenge positivity assumptions bounded away progressively increasing scenario estimators report table average bias bias multiplied standard error multiplied mean squared error mse multiplied across repetitions specifically realizations estimator based independent draws call standard error average bias mean squared error also represent series figures mse empirical coverage cis widths evolve sample size scenarios number covariates scenario parameter scenario increase ease comparisons similar figures share results scenarios increasing setting log results three simulation studies scenarios best presented commented upon altogether figure table summarize numerical findings scenario figure table summarize numerical findings scenario figure table summarize numerical findings scenario figures reveal general trend mse decreases sample size increases despite fact number covariates also increases different scenario overall perform similarly better tmle tmle outperforms iptw iptw outperforms moreover gap one hand tmle hand reduces sample size increases scenario reduces decreases sample size across scenarios judging tables unadjusted iptw estimators strongly biased tmle estimator strongly biased even large sample size number covariates sufficiently small note however bias tmle vanishes sample size scenario sample size action contrast estimators essentially unbiased configurations tables also reveal variance tmle estimator tends smaller estimators last two variances similar moreover gap tends diminish sample size increases scenarios turn figures estimator performs best terms empirical coverage followed tmle estimators order moderate sample size superiority cis others striking however even fail provide wished coverage except sample size large say larger side note recall drawn binomial law parameter probability approximately light empirical coverage abnormal independent cis exact coverage even moreover anticipated get conservative cis estimate asymptotic variance estimator see section fact ratio entries table scenario close one estimator sample size mention sample size table scenario reveals little asymptotic variance play finally see figures cis based estimators systematically slightly wider slightly narrower based tmle estimator much narrower based estimator scenario keeping fixed increasing figure table summarize numerical findings scenario number covariates set sample size goes steps observe trend figure figures mse decreases sample size increases overall perform similarly better tmle tmle outperforms iptw iptw outperforms moreover gap one hand tmle hand vanishes completely sample size increases whereas got smaller scenarios estimator mse tmle iptw sample size estimator ratio width coverage mse five seven estimators mse estimator large fit picture mse multiplied tmle sample size tmle estimator sample size coverage cis based doublerobust estimators relative width cis based estimators plain tmle figure scenario fix ratio increase sample size bias mse ratio bias mse ratio bias mse ratio table scenario performance estimator sample size ratio columns named correspond estimators respectively rows ratio report ratios average estimates across repetitions empirical bias multiplied mse multiplied estimator tmle mse iptw sample size estimator ratio width coverage mse five seven estimators tmle sample size tmle estimator sample size coverage cis based doublerobust estimators mse multiplied relative width cis based estimators plain tmle figure scenario increase sample size set bias mse ratio bias mse ratio bias mse ratio table scenario performance estimator sample size ratio columns named correspond estimators respectively rows ratio report ratios average estimates across repetitions empirical bias multiplied mse multiplied estimator tmle mse iptw sample size estimator ratio width coverage mse five seven estimators mse multiplied tmle sample size tmle estimator sample size coverage cis based doublerobust estimators relative width cis based estimators plain tmle figure scenario increase sample size keep log bias mse ratio bias mse ratio bias mse ratio table scenario performance estimator sample size log columns named correspond estimators respectively rows ratio report ratios average estimates across repetitions empirical bias multiplied mse multiplied judging table unadjusted iptw estimators strongly biased whereas estimators essentially unbiased even small sample size tmle estimator strongly biased sample size much less increases bias doublerobustness action moreover little difference tmle estimators terms bias mse sample size figure reveals sample sizes empirical coverage cis based estimators satisfactory cis based tmle estimator may provide coverage wished table ratio rows estimation actual variance tmle estimator quite good sample size apparently variance estimator sample size much better estimated sample size scenario keeping fixed increasing figure table summarize numerical findings scenario sample size set number covariates ranges take home message figure terms mse estimators outperform tmle estimator outperforms iptw estimators figure shows message also valid considering empirical coverage cis based different estimators number covariates increases empirical coverage degrade however cis based estimator behave remarkably better based estimator superior based tmle estimator examining table helps better understand general pattern unadjusted iptw estimators strongly biased compete tmle estimator performs rather well number covariates equals like estimators however tmle estimator biased compete even help yet moderate sample size contrast estimators essentially unbiased exhibit relatively small variances compared variances reported tables finally let note estimation variance estimator rather good see ratio rows table opposed variance estimator scenario keeping fixed challenging positivity assumption scenario study level posivitity violation influences performance estimators small sample size covariates progressively increasing figure illustrates positivity violation challenged recover fact increasing law highly skewed positivity assumption practically violated figures table summarize numerical findings scenario see figure overall tmle estimator much affected estimators near violation positivity assumption sample size estimators behave similarly terms mse empirical coverage judging table unadjusted iptw tmle estimators strongly biased compete estimator tmle mse iptw sample size estimator tmle sample size tmle estimator ratio width coverage mse five seven estimators mse multiplied sample size coverage cis based doublerobust estimators relative width cis based estimators plain tmle figure scenario fix number covariates increase sample size bias mse ratio bias mse ratio bias mse ratio table scenario performance estimator sample size columns named correspond estimators respectively rows ratio report ratios average estimates across repetitions empirical bias multiplied mse multiplied estimator tmle mse iptw number covariates estimator ratio width coverage mse five seven estimators mse multiplied tmle tmle estimator number covariates number covariates coverage cis based doublerobust estimators relative width cis based estimators plain tmle figure scenario fix sample size increase number covariates bias mse ratio bias mse ratio bias mse ratio table scenario performance estimator sample size columns named correspond estimators respectively rows ratio report ratios average estimates across repetitions empirical bias multiplied mse multiplied nearly unbiased estimators rather poor performance terms empirical coverage cis based estimators may due apparent failure estimating well variance see ratio rows table secondary simulation study procedure shorter section describe second less ambitious simulation study aim evaluate interest using procedure procedure specifically wish investigate context section rivals estimator also rely estimation iptw tmle estimators perform provided estimator indexed targeted parameter thus choose repeat independently times following steps number covariates simulate data set independent observations drawn derive estimator sections derive estimator section well competing iptw tmle estimators exactly presented section also using place others results reported table clear pattern emerges table bias systematically reduced using place nevertheless mse iptw estimator increases increase number covariates contrast estimator benefits substitution stark decrease mse top bias latter still far large makes remarkable fact tmle estimator greatly benefits substitution fronts bias mse contrary benefit estimator convincing summary targeted even context even used build plain tmle estimator opposed estimator discussion study inference value smooth statistical parameter law sample independent observations situations rely machine learning algorithm parameter estimate possibly consistently product rates convergence estimators components targets may slower convenient plain tmle chosen would typically lend construction selection would empirical bias something akin empirical variance estimator opposed tmle develop collaborative tmle procedure succeeds achieving relevant empirical processes conditions exists oracle makes bulky remainder term asymptotically gaussian asymptotically gaussian hence amenable building provided asymptotic variance estimated mse delta ecdf estimator tmle tmle tmle estimator estimator mse three seven estimators mse multiplied ratio width coverage every simulate observations compute finally plot corresponding empirical cumulative distribution delta propensity score delta delta coverage cis based doublerobust estimators relative width cis based estimators plain tmle figure scenario fix vary mses iptw large plot mses plain tmle two collaborative tmles ease comparisons bias mse ratio bias mse ratio table scenario performance estimator sample size columns named correspond estimators respectively rows ratio report ratios average estimates across repetitions empirical bias multiplied mse multiplied construction main result empirical behavior illustrated inference average treatment effect theoretically numerically simulation study estimated lasso bound norm candidate coefficients overall resulting estimator superior competitors including plain tmle estimator evaluated terms empirical bias standard error mean squared error coverage cis superiority striking small moderate sample sizes also strong number covariates increases positivity assumption increasingly challenged thus making inference task progressively even delicate simulation study suggests cis based provide wished coverage especially small sample sizes obviously may explained need estimator reach asymptotic regime subtly may also related highlevel assumption states existence oracle making bulky remainder term asymptotically gaussian assumption may fail hold practice devote future research understanding better finding strategies avoir relying conclusion believe present study demonstrates high versatility potential collaborative targeted minimum loss estimation methodology relative simplicity focused inference smooth statistical parameter independent identically distributed observations assuming machine learning algorithm finetuned parameter instantiation collaborative targeted minimum loss estimation methodology extended statistical parameters sampling schemes machine learning algorithms proofs sections respectively prove lemma theorem lemma corollary proof lemma proof hence also imply combined previous display yield showing satisfied proof theorem proof proof unfolds two parts step one extracting first order term equality rewrites second term sum first pair brackets first term second pair brackets entail satisfies also use fact depends thus holds let study side expression rewrites consider three terms side equation combining fact whenever reveals first third terms equal similar reasons second term equals therefore using successively obtain step two showing first order term complete rest proof consists showing term brackets say inequality follows definition triangle inequality equality therefore suffices prove absolute value side expression guaranteed every inequality yields hence convexity since get light side expression completes proof hence rewrites finally follows proof lemma proof set using inequality valid real numbers first remark therefore also holds second decompose difference lemma guarantees first term rhs expression uniformly bounded standard central limit theorem sequences independent identically distributed random variables finite variance implies second term finally third term inequality remark previously made summary stated proof corollary proof corollary uses repeatedly fact specific random functions fall donsker classes tending one specifically proof refer several times following lemma proof deferred end section lemma suppose assumptions corollary met tending one also fall classes present proof corollary proof corollary obvious counterpart appearing within yet consistently estimates converges limit may differ moreover since bounded away zero element model equal see proof lemma falls class tending one thus holds requested addition following convergence also occurs requested see proof consequently imply let turn assumption alleviate notation let second order derivative given definition see twice differentiable obviously exists universal constant supremum surely smaller consequently assumption met addition show hold true whenever met see assumed hold true summary satisfied construction met thus theorem applies implies result stated corollary completes proof proof suppose assumptions corollary met recall decomposition alleviate notation introduce lemma falls class tending one using repeatedly inequality obtain assumptions lemma met therefore retrieve bound proof assert conclusion holds completes proof proof suppose assumptions corollary met view therefore inequality equality yield completes proof proof context section suppose assumptions corollary met using inequality yields hence much spare observe evidently implies upper bound therefore furthermore lemma guarantees falls class tending one argument one lead section thus completes proof proof context section suppose assumptions corollary met view hence therefore inequality implies bound thus completing proof proof context section suppose assumptions corollary met lemma falls class probability tending one view holds argument one lead section thus completes proof proof lemma proceed order appearance statement lemma first note second derive explicit forms difference two last ones third recall explicit forms given derive difference thanks explicit forms straightforward applications theorem yield result conclude article final remark suppose met light also assume twice differentiable neighborhood assumptions implicit function theorem satisfied differentiable around references cochran effectiveness adjustment subclassification removing bias observational studies biometrics pages franklin eddings glynn schneeweiss regularized regression versus propensity score confounding adjustment secondary database analyses american journal epidemiology friedman hastie tibshirani regularization paths generalized linear models via coordinate descent journal statistical software url http gruber van der laan application collaborative targeted maximum likelihood estimation causal inference genomics international journal biostatistics article gruber van der laan ctmle package collaborative targeted maximum likelihood estimation technical report university california berkeley gruber logan monge ensemble learning inverse probability weights marginal structural modeling large observational datasets statistics medicine imai king stuart matchit matchit nonparametric preprocessing parametric casual inference package version pages imai king stuart matching nonparametric preprocessing reducing model dependence parametric causal inference political analysis horvitz thompson generalization sampling without replacement finite universe journal american statistical association combs lendle franklin wyss schneeweiss van der laan propensity score prediction electronic healthcare databases using super learner propensity score methods arxiv preprint gruber lendle chambaz franklin wyss schneeweiss van der laan scalable collaborative targeted learning data arxiv preprint schwab van der laan adaptive propensity score truncation causal inference arxiv preprint wyss franklin schneeweiss van der laan lasso constructing propensity estimators highdimensional data statistical methods medical research kurth walker glynn chan gaziano berger robins results multivariable logistic regression propensity matching propensity adjustment weighting conditions nonuniform effect american journal epidemiology lee lessler stuart improving propensity score weighting using machine learning statistics medicine core team language environment statistical computing foundation statistical computing vienna austria url https robins new approach causal inference mortality studies sustained exposure periodapplication control healthy worker survivor effect mathematical modelling robins robust estimation sequentially ignorable missing data causal inference models proceedings american statistical association section bayesian statistical science pages robins rotnitzky semiparametric efficiency multivariate regression models missing data journal american statistical association robins rotnitzky comment bickel kwon article inference semiparametric models questions answer statistica sinica robins hernan brumback marginal structural models causal inference epidemiology epidemiology robins rotnitzky van der laan comment profile likelihood murphy van der vaart journal american statistical association theory methods rosenbaum rubin central role propensity score observational studies causal effects biometrika pages rosenbaum rubin reducing bias observational studies using subclassification propensity score journal american statistical association schneeweiss rassen glynn avorn mogun brookhart highdimensional propensity score adjustment studies treatment effects using health care claims data epidemiology cambridge mass schnitzer cefalu collaborative targeted learning using regression shrinkage statistics medicine stitelman van der laan collaborative targeted maximum likelihood time event data international journal biostatistics article van der laan chambaz targeted learning data science causal inference complex longitudinal studies chapter ctmle continuous tuning springer series statistics springer van der laan rose targeted learning causal inference observational experimental data springer science business media van der laan rubin targeted maximum likelihood learning international journal biostatistics van der laan gruber collaborative double robust targeted maximum likelihood estimation international journal biostatistics article van der vaart asymptotic statistics volume cambridge series statistical probabilistic mathematics cambridge university press cambridge van der vaart wellner weak convergence empirical processes springer series statistics new york applications statistics wang rose van der laan finding quantitative trait loci genes collaborative targeted maximum likelihood learning statistics probability letters bias mse bias mse bias mse bias mse table using procedure performance estimator sample size prime symbol indicates use estimator place bias multiplied mse multiplied
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clustering implies geometry networks dmitri may northeastern university departments physics mathematics electrical computer engineering boston usa network models latent geometry used successfully many applications network science disciplines yet usually impossible tell given real network geometric meaning typical element ensemble random geometric graphs identify structural properties networks guarantee random graphs properties geometric specifically show random graphs expected degree clustering every node fixed constants equivalent random geometric graphs real line clustering sufficiently strong large numbers triangles homogeneously distributed across nodes real networks thus consequence network geometricity methods use prove quite general applicable network ensembles geometric certain problems quantum gravity equilibrium statistical mechanics often possible tell given system state typical state given ensemble network science statistical mechanics methods used successfully variety applications question often intractable stochastic network models define ensembles random graphs usually intractable distributions therefore usually unknown given real network typical element ensemble random graphs defined given model model appropriate real data yield reliable predictions progress made addressing problem classes models configuration stochastic block models interested network models models nodes assumed populate latent geometric space probability connections nodes usually decreasing function distance space models first introduced sociology model homophily social similar two people closer latent space likely connected since models used extensively many applications ranging predicting social behavior missing future links designing efficient information routing algorithms internet identifying connections brain critical function inferring community structure networks surveys simplest network model latent space model simplest latent space real line nodes points sprinkled randomly two nodes connected distance certain threshold random graph ensemble known gilbert model random geometric graphs even simplest model ensemble distribution intractable unknown therefore impossible tell given real network geometric typical element ensemble one always check simulations subset necessary conditions network geometric structural properties must match corresponding ensemble averages network property one usually means function adjacency matrix simplest examples functions numbers edges triangles subgraphs different sizes network distributions betweenness shortestpath lengths correspond much less trivial functions adjacency matrices since number propertyfunctions infinite since general intractable unknown impossible check properties match conditions necessary network geometricity satisfied sufficient conditions exist structural network properties random networks properties typical elements ensemble random geometric graphs answer question positively random geometric graphs show set sufficientcondition properties surprisingly simple properties expected numbers edges triangles equivalently expected degree clustering every node specifically consider ensemble random graphs expected degree every node fixed value expected number triangles every node belongs also fixed value seemingly nothing geometric ensemble since defined purely networkstructural triangles combination principle yet show clustering sufficiently strong ensemble equivalent ensemble random geometric graphs general ensemble sharp soft probability connections depending distance nodes larger smaller grand canonical probability function energies edges distances span strong clustering tally important property real networks thus appears consequence latent geometry simplest model networks strong clustering strauss model random graphs given expected numbers edges triangles strauss model well studied many problematic features including degeneracy phase transitions hysteresis caused statistical dependency edges constraints observed real networks particular strauss model triangles coalesce maximal clique portion nodes large degree clustering close rest nodes low degree zero clustering clustering organization differs drastically one real networks triangles homogeneously distributed across nodes modulo poisson fluctuations structural constraints want fix expected number edges triangles every node values strauss model fixed accomplish therefore instead begin canonical ensemble random graphs every edge occurs independently edges given probability pij general different different edges ensemble void pathologies expected degree hki number triangles hti node inpthe ensemble simply hki pij hti pij pjk pki connection probability matrix pij satisfying constraints hki hti yield canonical ensemble nodes expected degree number triangles however claim ensemble unbiased ensemble constraints particular matrix pij satisfying may enforce additional constraints expected values network properties words first find way sample matrices pij distribution subject desired constraints seemingly intractable problem finds solution using theory graph limits known graphons basic formalism introduced network models latent variables graphon symmetric integrable function essentially thermodynamic limit matrix pij fixed graph size graphon defines graph ensemble sprinkling nodes uniformly random interval connecting nodes probability pij sprinkled positions limit discrete node index becomes continuous graphs ensemble dense expected gree node interested sparse ensembles since real networks sparse average degrees either constant growing logarithmically network size model sparse networks one replace rescaled graphon depends expected degrees depend number triangles vanishes opposed clustering real networks depend size growing networks either solution impasse linearly growing support graphon let graphon whole infinite plane finite simply consider restriction finite square size graphon connection probability thermodynamic limit case expected degree number triangles node thermodynamic limit finite positive finite graph size graph ensemble defined sprinkling points uniformly random interval connecting nodes probability pij difference infinite graph ensemble thermodynamic limit latter case sprinkling realization poisson point process whole infinite real line main utility using graphons allow formalize task variational problem formulate first observe fixed sprinkling connection probability matrix pij also fixed since fixed pij edges independent bernoulli random variables albeit different success probabilities entropy graph ensemble fixed sprinkling pis sum entropies edges log log entropy bernoulli random variable success probability unfixing distribution entropy function random sprinkling ensemble known converge thermodynamic limit delta function centered graphon entropy defined gibbs entropy ensemble log bernoulli entropy thus large graph sampled typical representative ensemble proof dense graphons show appendix sparse settings well therefore sparse ensemble unbiased defined graphon maximizes graphon entropy subject constraints expected bers edges triangles every node fixed values find graphon maximizes entropy satisfies constraints observe constraint implies integrable since therefore first solve problem finite consider thermodynamic limit using method lagrange multipliers define lagrangian lagrange multipliers coupled degree triangle constraints equation leads following integral equation log appears intractable however inspired grand canonical formulation graph ensembles next show sufficiently large approximate solution following graphon energy distance nodes chemical potential inverse temperature functions rescaled inverse logarithm thermodynamic show first notice solution degree constraint becomes therefore average degree fixed depend log small last integral term expected number common neighbors nodes negligible simplifies equation graphs expected degree fixed solution constant integral longer negligible evaluate exactly exact expression fig rescaled number common neighbors versus different values terse omit brevity important property large closely approximated fig limit approximation becomes exact since heaviside step connected approximating commonneighbor integral noticing log transform equation solution solution consistent solution regime first value regimes second one check expected number common neighbors decays exponentially therefore common neighbor term indeed negligible regime even though prefactor large fixed large figure illustrates large expected average degree clustering ensemble functions respectively given values two constraints define two ensemble parameters solution eqs note large fig clustering close maximum computed analytically since approximations valid large apply graphs strong clustering sparse thermodynamic limit finite average degree chemical potential must finite must diverge temperature must zero graphs strongest clustering exact solution problem finite however graphs weaker clustering approximate solution emphasize fact graphon dependency via distance still homogeneously distributed across nodes albeit subject structural constraints imposed degree distribution shown random geometric graphs generalized satisfy additional constraint enforcing degree distribution generalization still uses grand canonical connection probability albeit hyperbolic geometry reproduces clustering organization real networks observations lead conjecture real networks typical elements ensembles soft random geometric graphs degree distribution constraints community structure another common feature real networks reflection node density latent geometry fig average degree clustering soft random geometric graphs connection probability functions dashed curves show simulation results averaged random graphs size interval periodic boundary conditions solid curves color corresponding analytic results using numeric evaluation color axes logarithmic scale color ticks evenly spaced log log approximate entropy maximizer means ensemble random graphs expected degree clustering every node fixed given constants approximately equivalent ensemble soft random geometric graphs specific form connection probability grand canonical distribution function maximizes ensemble entropy constrained fixed average energy number particles ensemble fermi particles graph edges edge pair nodes energy distance span average number particles fixed chemical potential fixing average energy fixing average number triangles equivalent smaller likely states larger thanks triangle inequality equivalence explains distribution appears approximate solution entropy maximization problem constrained fixed limit graphon becomes step function meaning soft random geometric graphs become traditional sharp random geometric graphs pair nodes connected approximations become exact limit degree distribution soft random geometric graphs poisson distribution many real networks power law triangles real networks final remark note methodology developed quite general applied network models latent variables geometric tell given model adequate given network also note similar class problems underlies approaches quantum gravity emerging geometry one expects continuous spacetime emerge classical limit fundamentally discrete physics planck scale perhaps directly related example hauptvermutung problem causal sets given lorentzian spacetime causal sets random geometric graphs edges connecting pairs events sprinkled randomly onto spacetime planck density continuous spacetime given begin discrete physics lead ensemble random graphs equivalent ensemble causal sets sprinkled onto spacetime observe answer question one solve ensemble equivalence problem solved except spacetime universe appendix show entropy considered sparse graph ensemble completeness first show average entropy density converges graphon entropy density thermodynamic limit show relative variance coefficient variation entropy distribution goes zero limit begin notations definitions notations definitions let interval length real numbers sampled uniformly random large binomial sampling approximates poisson point process unit rate since every uniformly distributed since independent probability density function sprinklings impose periodic boundary conditions making circle distance point xij distances xij uniformly distributed given ensemble ensemble graphs whose edges elements adjacency matrix aij independent bernoulli random variables abusing notation aij probability pij xij aij probability pij pii entropy random variable aij pij log log entropy bernoulli random variable success probability since aij independent ensemble entropy pij fixed given sprinkling ensemble ensemble graphs sampled first sampling random sprinkling sampling random graph consider entropy random variable defined random variable relative variance vanishes thermodynamic limit hsn hsn hsn stands averaging across random sprinklings first show average entropy average entropy per node hsn graphon entropy density hsn lim xij dxij xij dxij dxi dxj average ensemble entropy density converges graphon entropy density using definitions observations get hsn dxk inn hsn dxk inn dxk integration variables indices equal either yields factor hsn dxi dxj also swapped summation integration changing variables xij integrating yields another factor hsn xij dxij since xij uniformly distributed terms sum contribute equally bringing another factor total number terms sum hsn xij dxij thus hsn lim ensemble entropy compute hsn hsn must calculate dxm inn dxm inn prove notational convenience equations extended support xij xij first integral different instead therefore immediately conclude converges finite limit bianconi pin marsili proc natl acad sci usa anand bianconi phys rev calculate second integral use anand krioukov bianconi phys rev integrate variables indices garlaschelli loffredo phys rev equal bringing factor zzzz garlaschelli loffredo phys rev lett dxi dxj dxk dxl squartini mastrandrea garlaschelli new phys changing variables xij peixoto phys rev xkl integrating thus bringing peixoto phys rev lett peixoto phys rev another factor get newman peixoto phys rev lett xij xkl dxij dxkl mcfarland brown bonds pluralism form substance urban social networks john wiley new york since xij xkl independent uniformly dis mcpherson cook annu tributed every term double sum conrev sociol tributes equally total number terms hoff raftery handcock stat assoc yielding sarkar chakrabarti moore ijcai xij xkl dxij dxkl tita brantingham galstyan cho discret contin dyn syst ser papadopoulos krioukov nat commun oukov nat commun hsn phys rep bianconi epl collecting calculations hsn finally gilbert soc ind appl math obtain penrose random geometric graphs oxford university press oxford hsn orsini dankulov makovic mahadevan vahdat bassler hsn toroczkai caldarelli fortunato krioukov nat commun czabarka toroczkai phys rev lett squartini mol den hollander garn laschelli phys rev lett dettmann georgiou phys rev penrose ann appl probab radicchi castellano cecconi loreto thank lippner topalov piskunov parisi proc natl acad sci kitsak toroczkai radicchi castellano phys rev baryshnikov useful discussions suggestions work supported nsf strauss siam rev foster foster paczuski grassberger phys rev park newman phys rev albert rev mod phys radin ren sadun phys math theor park newman phys rev serrano sci rep gao barzel nature garlaschelli caldarelli epl bianconi eur lett large networks graph limits american mathematical society providence caldarelli capocci rios phys rev lett phys rev boccaletti latora moreno chavez hwanga phys rep borgs chayes cohn zhao janson nyjm monogr garlaschelli ahnert fink darelli entropy serrano krioukov phys rev lett krioukov papadopoulos kitsak vahdat phys rev zuev bianconi krioukov sci rep menichetti rahmede bianconi sci rep sorkin lectures quantum gravity edited gomberoff marol springer new york bombelli lee meyer sorkin phys rev lett
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consensus neural networks chinese reading comprehension yiming ting zhipeng shijin guoping iflytek research beijing china research center social computing information retrieval harbin institute technology harbin china ymcui zpchen gphu tliu mar abstract reading comprehension embraced booming recent nlp research several institutes released reading comprehension data greatly accelerated research machine comprehension work firstly present chinese reading comprehension datasets consist people daily news dataset children fairy tale cft dataset also propose consensus neural network architecture tackle reading comprehension problem aims induce consensus attention every words query experimental results show proposed neural network significantly outperforms baselines several public datasets furthermore setup baseline chinese reading comprehension task hopefully would speed process future research introduction ultimate goal machine intelligence read comprehend human languages among various machine comprehension tasks recent research reading comprehension task attracted lots researchers reading comprehension problem taylor aims comprehend given context document answer questions based nature document answer single word document thus reading comprehension described triple document query answer query adopting neural network approaches bahdanau machine able learn relationships document query answer known neural network based approaches need training data train reliable model predictions hermann published mail news corpus reading comprehensions content formed news articles summarization also hill released children book test cbt corpus research training samples generated automatic approaches see automatically generating training data neural network training essential reading comprehension furthermore difficult problems reasoning summarization context need much data learn interactions though seen many improvements public datasets researchers suggested dataset requires less inference expected chen furthermore public datasets automatically generated indicate pattern training testing phase nearly easier machine learn patterns paper release chinese reading comprehension datasets including people daily news datasets children fairy tale datasets highlight datasets human evaluated work done joint laboratory hit iflytek hfl work licensed creative commons attribution international licence http licence details dataset testing purpose harder machine answer questions automatically generated questions human evaluated dataset processed may accordance pattern automatic questions detailed analysis given following sections main contributions paper follows knowledge first released chinese reading comprehension datasets human evaluated test sets benefit research communities reading comprehension also propose refined neural network aims utilize full representations query deal reading comprehension task model outperform various baseline systems public datasets rest paper organized follows section briefly introduce existing datasets describe chinese reading comprehension datasets detail section show refined neural network architecture reading comprehension experimental results public datasets well chinese reading comprehension datasets given section related work described section make brief conclusion work end paper chinese reading comprehension datasets first begin brief introduction existing reading comprehension datasets introduce chinese reading comprehension datasets people daily children fairy tale existing datasets typically two main genres datasets publicly available stem english reading materials news articles often come short summary whole report spirit hermann constructed large dataset cnn daily mail news data firstly regard main body news article document query formed summary article one entity word replaced placeholder indicate missing word finally replaced entity word answer query also proposed anonymize named entity tokens data entity tokens every sample order exploit general relationships anonymized named entities rather common knowledge chen studies datasets showed anonymization less useful expected children book test also dataset called children book test cbt released hill built children book story different previously published mail datasets formed document consecutive sentences book regard sentence query one word blanked placeholder missing word chosen named entities common nouns verbs prepositions verbs prepositions less dependent document studies focusing datasets people daily children fairy tale datasets part introduce chinese reading comprehension datasets though many solid works previously described public datasets studies chinese reading comprehension datasets makes datasets different previous works listed far know proposed dataset first chinese reading comprehension datasets add language diversity community cnn daily mail datasets available http cbt datasets available http datasets available http document query answer people daily jan according report new york times wall street stock market continued rise global stock market last day ending highest record near record year new york times reported index rose year largest increase since dow jones industrial average index rose largest increase since nasdaq rose terms december due prospects employment possible acceleration economy next year rising confidence consumers reported business association report consumer confidence rose december significantly higher november also wall street journal reported best stock market since year chase silly money wise way invest stock silly money buy hold common combination stock strategy better complex investment methods hedge funds methods adopted professional investors silly money xxxxx buy hold common combination stock strategy figure example training sample people daily datasets english translation given right box xxxxx represents missing word example document consists sentences sentence chosen query provide chinese reading comprehension data news domain well validation test data test release two test sets deserves highlight one test sets made humans makes harder answer automatically generated test set people daily roughly collected news articles people daily following liu process news articles triple form detailed procedures follows given certain document composed set sentences randomly choose answer word document note restrict answer word noun well answer word appear least twice document sentence segmentation identified using ltp toolkit che distinguish named entities common nouns hill second answer word chosen sentence contains defined query answer word replaced specific placeholder hxi third given query document target prediction recover answer way generate tremendous triples training proposed neural network without assumptions nature original corpus note unlike previous work using method mentioned document different queries makes general generate training data neural network training figure shows example people daily datasets children fairy tale except validation test set people daily news data also present two test sets well two test sets made children fairy tale cft fairly different news genre reason set test sets children fairy tale mainly consists stories animals virtualized characters http people daily train valid test query max tokens docs max tokens query avg tokens docs avg tokens query vocabulary children fairy tale table statistics people daily datasets children fairy tale datasets prevents utilizing gender information world knowledge training data important solving several types questions coreference resolutions etc cft dataset one test set automatically generated using algorithms described one made human suggest latter harder former one automatically generated test sets aware fixed collocation words thus pattern around query blank exactly appeared document much easier machine identify correct answer building human evaluation test set eliminated types samples makes harder machine comprehend intuitively human evaluation test set harder previously published test sets statistics people daily news datasets well children fairy tale datasets listed table consensus attention sum reader section introduce neural network model reading comprehension task namely consensus attention sum reader cas reader model primarily motivated kadlec aims directly estimate answer document instead making prediction full vocabularies noticed concatenating final representations query rnn states enough representing whole information query propose utilize every time slices query make consensus attention among different steps formally given set training triple construct network following way first convert representation document query continuous representations shared embedding matrix query typically shorter document sharing embedding weights query representation benefited embedding learning document side better separating embedding matrices individually use two different rnns get contextual representations document query capture contextual information history future implementation use gated recurrent unit gru modeling cho gru gru take hdoc hquery represent contextual representations document query tensor shape directly make dot product hdoc hquery get importance document word respect query word time sum attention layer merging function individual attention layer layer embedding layer mary sits beside says love mary document loves blank query figure architecture proposed consensus attention sum reader cas reader use softmax function get probability distribution document hdoc also known attention sof tmax hdoc hquery way every time step query get probability distribution document denoted means attention value ith word document time length document get consensus attention individual attentions explicitly define merging function denote final attention document length query paper define merging function one three heuristics shown equations sof tmax mode sum sof tmax mode avg tmax max mode max finally map attention result vocabulary space sum attention value occurs different place document shares word kadlec indicate position word appear document figure shows proposed neural network architecture experiments experimental setups training details neural network models illustrated follows embed units hidden units dropout none none none cnn news cbtest cbtest people daily cft table neural network setups task note dropout applied output grus cnn news train valid query max candidates avg candidates avg tokens vocabulary test cbt train valid test cbt train valid test table statistics public reading comprehension datasets cnn news data cbtest named entites common nouns embedding layer use randomly initialized embedding matrix uniformed distribution interval note word embeddings used experiments hidden layer initialized gru units random orthogonal matrices saxe gru still suffers gradient exploding problem set gradient clipping threshold experiments pascanu vocabulary training efficiency generalization people daily cft datasets truncate full vocabulary set shortlist unknown words mapped different specific symbols using method proposed liu vocabulary truncation cnn cbtest dataset optimization used adam update rule kingma initial learning rate used negative training objective function batch size set neural network setups dimensions embedding layer hidden layer dropout srivastava task listed table trained model several epochs choose best model according performance validation set models trained tesla gpu model implemented theano theano development team keras chollet results public datasets verify effectiveness proposed model first tested model public datasets evaluation carried cnn news datasets hermann cbtest datasets hill statistics datasets listed table done datasets experimental results given table evaluate model terms accuracy due time limitations evaluate model ensemble cnn news performance cnn news datasets shows model par attention sum reader decrease validation improvements test set failed outperform stanford model stanford utilized glove embeddings pennington deep lstm attentive impatient human lstms memnn window stanford reader cas reader mode avg cnn news valid test cbtest valid test cbtest valid test table results cnn news cbtest named entity common noun datasets results marked taken hermann taken hill taken chen taken kadlec people daily valid test reader cas reader mode avg cas reader mode sum cas reader mode max children fairy tale table results people daily datasets children fairy tale cft datasets normalized probabilities named entities document rather words could make difference results model optimize certain type dataset make general cbtest cbtest dataset model gives slight improvements reader improvements validation set improvements test set cbtest though slight drop validation set declines boost test set absolute improvements suggest model effective beneficial consider every slices query answering results chinese reading comprehension datasets results chinese reading comprehension datasets listed table see proposed cas reader significantly outperform reader types test set maximum improvements cft dataset results indicate making consensus attention multiple time steps better relying single attention reader similar use model ensemble also consensus voting result different models also evaluated different merging functions results see avg sum methods significantly outperform max heuristics max heuristics failed outperform reader possible reason explained max operation sensitive noise word given high probability one time step query avg sum could easily diminish noise time steps higher value given word max situation noise removed preserve till end final attentions influence predictions lot also noticed though achieved accuracy among people daily datasets significant drop two cft test sets furthermore human evaluated test set meets sharp decline accuracy automatically generated test set analyses concluded regard cft datasets tests gap training data cft test data poses declines test sets problems remedied introducing similar genre training data regardless absolute accuracies cft datasets human test set much harder machine read comprehend discussed results see big gap automatically generated queries questions note test set query also formulated original sentence document suggest use general form queries another rise comprehension difficulties example instead asking went xxxxx morning change general question form morning makes harder machine comprehend gap general question form training data related work many reading comprehension models proposed models indicate attention mechanism essential machine comprehensions hermann proposed methodology obtaining large quantities triples using method large number training data obtained without much human intervention make possible train reliable neural network study inner relationships inside triples used neural networks task evaluation datasets showed approach effective traditional baselines hill also proposed similar approach large scale training data collections children book reading comprehension task using memory network heuristics surpass methods predicting named entities common nouns cbt cnn benchmark cas reader closely related work kadlec proposed use simple model using attention result directly pick answer document rather computing weighted sum representation document using attention weights like previous works proposed model typically motivated pointer network vinyals model aims solve one particular task answer single word appear document least experimental results show model outperforms previously proposed models large margin public datasets cbtest datasets liu proposed effective way generate exploit pseudo training data zero pronoun resolution task main idea behind approach automatically generate largescale pseudo training data using neural network model resolve zero pronouns also propose training phase adaptation phase also applied tasks well experimental results ontonotes corpus encouraging proposed approach significantly outperforms methods work proposed entirely new chinese reading comprehension dataset add diversity existing reading comprehension datasets moreover propose refined neural network model called consensus sum reader though many impressive progress made public datasets believe current machine comprehensions still stage discussed previous section answer pseudo query document enough machine comprehension general question form seen comprehensive processing human brains though test set still somewhat easy machine comprehend harder automatically generated test set releasing dataset let move step forward questions becomes good bridge automatic questions questions conclusion paper introduce first chinese reading comprehension datasets people daily children fairy tale furthermore also propose neural network model handle reading comprehension problems model able take question words accounts computing attentions document among many public datasets model could give significant improvements various baselines also set baseline chinese reading comprehension datasets hopefully make starter future studies future work carried following aspects first would like work another dataset contain questions far difficult existing datasets publicly available second going investigate hybrid reading comprehension models tackle problems rely comprehensive induction several sentences acknowledgements would like thank anonymous reviewers thorough reviewing proposing thoughtful comments improve paper work supported national leading technology research project via grant key projects national natural science foundation china via grant national natural science youth foundation china via grant references dzmitry bahdanau kyunghyun cho yoshua bengio neural machine translation jointly learning align translate arxiv preprint wanxiang che zhenghua ting liu ltp chinese language technology platform proceedings international conference computational linguistics demonstrations pages association computational linguistics danqi chen jason bolton christopher manning thorough examination mail reading comprehension task association computational linguistics acl kyunghyun cho bart van merrienboer caglar gulcehre dzmitry bahdanau fethi bougares holger schwenk yoshua bengio learning phrase representations using rnn statistical machine translation proceedings conference empirical methods natural language processing emnlp pages association computational linguistics chollet keras https karl moritz hermann tomas kocisky edward grefenstette lasse espeholt kay mustafa suleyman phil blunsom teaching machines read comprehend advances neural information processing systems pages felix hill antoine bordes sumit chopra jason weston goldilocks principle reading children books explicit memory representations arxiv preprint rudolf kadlec martin schmid ondrej bajgar jan kleindienst text understanding attention sum reader network arxiv preprint diederik kingma jimmy adam method stochastic optimization arxiv preprint ting liu yiming cui qingyu yin shijin wang weinan zhang guoping generating exploiting pseudo training data zero pronoun resolution arxiv preprint razvan pascanu tomas mikolov yoshua bengio difficulty training recurrent neural networks icml jeffrey pennington richard socher christopher manning glove global vectors word representation proceedings conference empirical methods natural language processing emnlp pages association computational linguistics andrew saxe james mcclelland surya ganguli exact solutions nonlinear dynamics learning deep linear neural networks arxiv preprint nitish srivastava geoffrey hinton alex krizhevsky ilya sutskever ruslan salakhutdinov dropout simple way prevent neural networks overfitting journal machine learning research wilson taylor cloze procedure new tool measuring readability journalism mass communication quarterly theano development team theano python framework fast computation mathematical expressions arxiv may oriol vinyals meire fortunato navdeep jaitly pointer networks advances neural information processing systems pages
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dna constructing boxes vijay arxiv dec department computer science northwestern university evanston usa kao department computer science yale university new usa vijayr abstract propose mathematical model dna using tiles form nanostructures first work combine studies nanotechnology rothemund winfree case model precise superset tile assembly model facilitates building scalable molecules model present algorithms build hollow cube intuitively one simplest structures construct also introduce five basic measures complexity analyze algorithms model algorithmic techniques applicable complex nanostructures introduction dna nanotechnology dna two related technologies enormous potentials goal dna nanotechnology construct small objects high precision seeman visionary work pioneered molecular units used objects decade later molecules proposed seeman molecules labean dna building blocks laboratory efforts successful generating interesting molecular structures including small cube chen seeman however immutable limited size mainly fabrication based mathematical model extended necessary parallel dna nanotechnology studies dna tiles focused using local deterministic binding rules perform computations rules based interactions exposed dna sequences individual tiles tiles assemble particular structure solution encoding computation winfree formulated model computations using molecules winfree used tiles computations constructions molecules labean first compute molecules supported part nsf grants part work performed author visiting department computer science yale university new usa supported national defense science engineering graduate fellowship combining two technologies several researchers demonstrated power dna nanostructure fabrication winfree investigated use molecules build lattice dna crystals rothemund winfree proposed mathematical model complexity measure building structures natural extension seminal results winfree rothemund winfree would creation nanostructures using tiling initiate extension paper proposes general mathematical model constructing structures tiles identifies set biological algorithmic issues basic implementation model provides basic computational concepts techniques address issues model paper focuses problem constructing hollow cube intuitively one simplest structures construct present algorithms problem analyze terms five basic measures complexity three natural approaches building hollow cube first approach uses tiles form tiles uses tiles construct cube paper fully investigate possibility incovenient shape molecules see sect algorithms modified accommodate tiles second approach builds cube genuine tiles focus paper third approach perhaps natural build cube genuine tiles yet clear tiles could created conceivably cube chen seeman may lead tiles form paper fully investigate possibility either approach algorithmically straightforward similar case basic idea algorithms use tiles form shape plane fold box illustrated fig easily synthesize set tiles create intitial shape overcome negligible probability success due biochemical factors must put many copies tiles solution must worry multiple copies shape interfering preventing folding figure avoid problem introduce randomization different copies shape unique sticky ends growth tiles complete structure must still deterministic based hybridization randomize computation input seed tiles rest shape assembles edges still relate depend random input different shape solution input form low probability interference another copy shape kept minimum raises another important issue using communicate information one part shape another since edges must relate random input designing local rules becomes nontrivial paper explore formalize patterns used completing task addition formalize biological steps allow specific subset fig planar shape fold box section formed many smaller dna tiles edges number complementary sticky ends exposed hybridize folding shape box edges hybridized hybridization edges whose two complements close proximity cause edge hybridize form complete box multiple copies shape solution copies shape interfere attach infinitely without control long edges matching sticky ends tiles added isolated period time thus allowing better control growth couple use temperature improve probability successful construction remainder paper organized follows section describes model computation including notation dna tiles definitions complexity measures section describes algorithms detail section discusses future research possibilities model computation section formally introduce model generalized tile assembly model mathematical biological level extension model presented rothemund winfree molecular units begin biological foundation model intend build structures using folding technique shown fig allow construction structures possible tile assembly model model relies using molecular building block dna tile tiles naturally hybridize form stable shapes varying sizes individual tiles easily customized replicated via synthesis pcr procedure specific algorithm dna tiles small nucleotides exposed action sites also known sticky ends dna strand consisting sequence base pairs sequence matches complementary sequence action site another tile hybridization property dna causes two molecules bind together forming larger structure tile synthesized laboratory specific sticky ends different combinations sticky ends tile essentially yield puzzle pieces tiles automatically hybridize left solution work uses molecules tiles shape molecules causes problem construction since sticky ends diagonally opposite ends see tiles form structures ragged edges hybridize figure algorithms easily modified use tiles adjusting proper alignment folding box sticky end sticky end individual molecules fig structure formed molecules left right sides hybridize aligned improperly true top bottom sides dna tile snynthetic dna tile derived structure trna however propose simpler alternative using branched molecules seeman variant derived structure trna molecules sketched figures truly sticky ends four sides structure stable sticky ends solution molecules flexibility align properly folding molecules offer natural motivation modeling using wang theory tiling allows abstract construction using molecules symbolic level symbolic representation tiles definition dna sequence length ordered sequence base pairs sequence end set base pairs assume directions explicity written sequence written direction complement sequence denoted define concatenation two sequences denoted simply sequence subsequence sequence denoted sequence given definitions two dna strands hybridize complementary sequences formally hybridize exist integers assume misbindings condition must met exactly errors binding remark note rather definition threshold temperature dna sequence temperature fixed set sequence unable remain stably hybridized complement solution temperature higher heating solution generally denatures strands definition strong biological foundation consequences methodology using temperature designing dna sequences tiles discussed lower threshold temperature say binds weaker work dna computing model uses dna sequences encode case identifier specifying kinds matches allowed tiles given side since misbindings identifiers map uniquely dna sequences present sides tiles bind formally following definition let set symbols used represent patterns sides tiles assume closed complementation exists complement purpose clear let set dna sequences called dna words words interfere bind inappropriately define injective map enc encoding symbol dna word map obeys complementation enc enc assumption sequences written without directions given means complement standard convention technically correct fixed set threshold temperatures simplifies model corresponds temperature parameter compensate allow actual threshold temperature deviate slightly fixed point condon corn marathe done work designing good dna sequences problems like one definition dna tile symbols enc exposed dna sequence north east south west action site tile given two tiles bind two sides complementary symbols properties hybridization including threshold temperature carry hybridization tiles make stronger assumption tiles requiring sticky ends tiles match exactly fully stage model exactly matches rothemund winfree except tiles rotate corresponds closely tile structure model point could require many symbols express different tile types possibly exponential number dna words ideally would like arbitrarily extend symbolic informational content side tile therefore make following generalization definition let set symbols closed complementation let set corresponding dna words generalization model defines map corresponding encoding genc genc enc abstract tile definition symbols equivalent dna tile define complementation follows let genc enc enc enc makes hybridization condition equivalent complementary symbols sides bind definition purposefully broad order allow different algorithms define encoding based number words tiles needed generalization genc enc original tile assembly model paper use following model definition concatenation generalization generalization maps every combination symbols unique symbol partition set contains complement symbols define genc enc enc let enc genc genc words concatenation generalization model straightforward extension tile model side tile corresponds symbols dna sequence corresponding action site simply map enc defined done since dna sequence corresponding symbol unique complement therefore unique complement symbol concatenation encodings individual using simple model reduce number dna words needed create simpler descriptions tiles algorithmic procedures models tiles discuss procedures growing larger structures follow rothemund winfree markov use common assumption structure begins seed tile grows timestep hybridization another new tile hybridizes given position following one two types rules deterministic given surrounding tiles position one tile type specific sticky ends sides fit randomized multiple tile types different sticky ends sides could fit position given tiles present new action site created probability proportional concentration tile type solution therefore grow structure algorithm repeats steps structure complete add tiles solution wait adhere growing structure optionally removes excess tiles solution washing cycling temperature steps prevent induce binding based threshold temperatures done waiting hybridization called binding complexity consider five basic methods analyzing algorithms using model time complexity algorithm sequence steps described thus natural measure time complexity model number steps required describes laboratory time space complexity number distinct physical tile types actual number molecules produced space complexity introduced describes amount unique dna synthesis necessary alphabet size number dna words rough laboratory limit size symbol set used corresponds directly number words practical significance potentially concatenation model could cause interference among tiles maintain misbindings however model removes analysis addition theoretically possible design dna words interference occur depending algorithm reality multiple tiles hybridize structures consisting one tile hybridize lose generality markov assumption generalization level generalization level amount information side tile related length sticky ends thus biological consequences number actual dna words via probability misformation misformed structures contain tiles bound properly sides assuming markov model consider adding tile partial structure complete hybridization requires binding two sides manages hybridize one side action site match misformation quantify probability following definition let success probability step probability tile solution binds possible sides partial structure given spot step addition tile spot structure resulting correct correct ncorrect nall ncorrect number tile types correctly bind nall number tile types solution could bind possibly even incompletely call misformation probability step algorithm additions misformation probability algorithm algorithm misformationproof misformation probability every step zero yielding zero total probability misformation hollow cube algorithms section examine algorithms designed use model build hollow cube using folding technique shown fig let length side cube input algorithms present interesting algorithm detail discuss others briefly overview figure illustrates planar shape algorithms construct reference labels shading regions figure discussion stated earlier sect must make shape unique different partial structures solution bind interfere unique seed structure use basic rules make edges shape correspond folding occur three basic patterns used construct different parts shape random assembly implements random rule see sect formally add tiles set distinct tiles equal concentrations could base strip growth pattern pattern pattern growth pattern arrow shows growth direction fig regions planar shape pattern pattern potentially hybridize completely given position thus information position completely random tiles differ component exposed assuming generalization straight copy see figure tiles added copy pattern along one end region parallel end adjacent region constructed rule deterministic turn copy see figure tiles added copy pattern along one end region perpendicular end adjacent region constructed counters required position tiles appropriately complete deterministic rule algorithm begins assembling random pattern string copied top bottom box random patterns added remaining edges box copied corresponding edges accordingly refer fig corresponding edges finally shape fold cut raising temperature assuming bonds tiles along weak threshold temperatures regions shaded fig cut away notation algorithms use concatenation generalization model thus tile triplets write change order tuple easily identify tiles bind since binding decision arbitrary purely notational assume tiles oriented directions clear define set random patterns used components exposed triplets tiles used random assembly use become clear discuss implementation random assembly use counters control growth planar shape concept well explored earlier papers tile assigned position plane denoted horizontal vertical coordinate create symbols position counters allow tiles hybridize positions match creating mechanism algorithms place tiles absolute relative positions let symbols denoting horizontal position vertical position respectively algorithm algorithm sacrifices time complexity reduce space complexity alphabet size addition using steps binding algorithm eliminates possibility misformations implementation random assembly base strip shown fig created via random assembly represents unique seed structure shape pattern assembled strip copied edges strip length edge cube use horizontal counters control growth add following tiles solution tiles vary since ends given tiles bordering ends given pattern tiles middle strip given appropriate horizontal counter markings every position every base strip must appropriate ends length note pattern exposed north south side tiles position given completely random one tiles containing could hybridize thus shape unique identifier namely sequence patterns along base strip probability sequence expressed given shape implementation straight copy must complete base strip regions addition patterns required elsewhere done using steps tiles done parallel following example assume triplet constant following tiles added step tiles versions exist patterns present tile description assume binds weaker cycling temperature ensures tiles attached side tiles added sequentially prevent misformations evident borders regions given symbols exposed sides hybridize strip tiles form folding edge box middle tiles copy random pattern another location steps middle regions complete except two rows top rows one row bottom order rows important add extra set tiles prevent top bottom hybridizing folding complete implementation turn copy step example copying bottom edge left edge shape fold done using vertical horizontal counters essentially places tile specific spot therefore add tiles complete region without possibility misformation example region would add following tiles let vary add copies pattern following adds left edge extra precaution set binding weaker cycle temperature several times addition force encoding horizontal vertical counters edges folding occurs sticky ends fact complementary summary random assembly base strip use straight copy using counter regions copy base strip pattern use straight copy fill bodies regions add edges using random assembly correspond portions shape use turn copy make edges correspond finally sequence straight turn copies complete shape raise temperature cut away shaded regions analysis algorithm theorem proof symbol set consists following pattern symbols horizontal vertical counters positions throughout shape specialized symbols dependent inputs theorem algorithm time complexity approximately proof algorithm consists following steps step build base strip steps complete steps grow step add random patterns leaf borders step first set steps steps complete regions time complexity basic version therefore approximately steps theorem space complexity algorithm approximately tiles proof clear counting number tiles required including necessary variations tile position pattern theorem number distinct temperatures required proof one temperature required detaching excess portions shape highest temperature level main portion box remains intact adding tiles rows involves cycling temperature prevent misformations ensuring side tiles bind requires third temperature two discussed potential misformations denature theorem misformation probability proof random assembly steps grow shape one direction using onedimensional counter correct tiles marked appropriate counters thus probability steps zero steps use vertical horizontal counters defining position bind weaker random patterns three words tiles pattern also misformation probability zero finally steps performed sequence potential spots misformation sealed tiles one row time probability misformation also zero thus misformation probability algorithm algorithm algorithm take full advantage parallelism instead adding one row another synthesize tiles horizontal vertical counters tile type fit one absolute position begin virtually tiles solution allow assembly proceed constant time random assembly performed special counters control length however strips must assembled separately independently prevent interference sequences exposed opposite random patterns dictate strips later hybridize larger shape random patterns cause growth makes shape unique needed turn copy implementation unchanged counters already present straight copy rather using constant symbol must instead use counters growth directions although algorithm straightforward tiles solution might get incomplete partial structures specifically tiles bind sides counters component thus creating shapes information allowing fold case positions misformation could occur counters bind patterns misformation probability prevent requiring counters bind weaker cycle temperature requiring rather distinct temperature levels giving misformation probability practically little meaning since may still get incomplete partial structures may wait long time proper hybridization occur besides drawback algorithm performs well complexity measures since counters necessary algorithm alphabet size greatly increased algorithm space complexity essentially different tile type created every position every pattern could present algorithms considered several algorithms using generalization model experimenting different uses omissions counters implementations patterns algorithm keeps space complexity time complexity low removing counters controlling growth certain rows columns region probability misformation increased algorithm highest misformation probability although low time space complexity algorithm frame regions constructed first possible filler tiles added strengthen structure later potential stability problems shape growth finally another possibility build six faces separately allow hybridize method little control final shape actually forms solution conclusion paper introduces precise extension tile assembly model allows greater information content per tile scalability three dimensions model better formalizes abstraction dna tiles symbols introduces five complexity measures analyze algorithms first extend nanostructure fabrication three dimensions addition paper opens avenues research first may possible encode information tiles succintly algorithms accomplish copy patterns discussed existence good generalization algorithms unknown algorithms form structures various applications biology computation studied work also must done quantify probabilities specified paper possibly including analysis tile binding finally remain important biological issues particular design strong tile suitable method computation design building block two important steps increasing feasibility use temperature may refined exploited improve complexity results number steps needed lab references chen seeman synthesis dna molecule connectivity cube nature condon corn marathe combinatorial dna word design winfree gifford seeman dna molecules biochemistry labean winfree reif experimental progress computation dna tilings winfree gifford labean yan kopatsch liu winfree reif seeman construction analysis ligation dna triple crossover complexes chem markov crystal growth beginners fundamentals nucleation crystal growth epitaxy world scientific singapore rothemund winfree complexity squares yao editor proceedings annual acm symposium theory computing portland may acm special interest group algorithms computation theory seeman junctions lattices journal theoretical biology wang proving theorems pattern recognition bell system technical journal winfree computational power dna annealing ligation baum lipton editors dna based computers dimacs series discrete mathematics theoretical computer science pages american mathematical society may winfree eng rozenberg string tile models dna computing condon rozenberg editors dna based computers leiden netherlands june leiden center natural computing winfree gifford editors preliminary proceedings fifth international meeting dna based computers cambridge massachusetts june dimacs winfree liu wenzler seeman design dna crystals nature
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context generation formal specifications analysis tools michele julien trustinsoft paris france cea list software reliability security laboratory cedex france sep abstract analysis tools like abstract interpreters symbolic execution tools testing tools usually require proper context give useful results analyzing particular function context initializes function parameters global variables comply function requirements however may write hand handwritten context might contain bugs match intended specification robust approach specify context dedicated specification language hold analysis tools support properly may mean put significant development efforts enhancing tools something often feasible ever possible paper presents way systematically generate context formal specification function applied subset acsl specification language order generate suitable contexts abstract interpretationbased value analysis framework analysis code written idea presented implemented new plugin currently use operational industrial setting keywords formal specification code generation transformation code analysis acsl introduction code analysis tools nowadays effective enough able provide suitable results code nevertheless several tools including abstract interpreters symbolic execution tools testing tools must analyze whole application program entry point main function else either executed provide imprecise results unfortunately entry point necessarily exist particularly analyzing libraries case verification engineer must manually write context analyzed function main function initializes parameters well necessary global variables mandatory initialization step must enforce function requirements may restrict possible input values sake memory footprint time efficiency analysis approach however work done first author cea list software reliability security laboratory additionally usual pitfalls software development bugs code maintenance etc handwritten context may match function requirements restrictive moreover kind shortcomings may difficult detect due fact context explicitly verification objective valid robust alternative specify context dedicated specification language make analysis tools handle properly often arduous approach support particular specification language feature may entail significant development process something often feasible ever possible also requires every tool paper presents way systematically generate analysis context formal specification function function requirements well additional restrictions input domains expressed function preconditions specification language short acsl specification interpreted constraint system simplified much possible converted code exactly implements specification indeed every possible execution satisfies conversely execution every possible input satisfying constraints expressed present formalization idea expressive subset acsl including standard logic operators integer arithmetic arrays pointers pointer arithmetic predicates validity initialization properties memory location ranges also provide implementation details tool named cfp context preconditions implemented code analysis framework code written thanks aforementioned technique cfp generates suitable contexts two abstract value analysis tools namely eva trustinsoft company tools actually distinct evolved versions older called value particular trustinsoft successfully used cfp library also known polarssl open source implementation building verification kit worth noting cfp revealed mistakes contexts previously written hand expert verification engineers comparing results pieces code also cfp generates code close possible code quite readable follows code patterns experts tools manually write contributions contributions paper threefold novel technique systematically generate analysis context formal specification function precise formalization technique presentation tool implementing technique used operational industrial setting outline section presents overview technique motivating example section details preconditions constraints conversion section explains code generation scheme latter section evaluates approach section discusses related work section concludes work also discussing future work https iii overview motivating example illustrate approach context generation function aes crypt cbc cryptographic utility implemented library figure shows prototype acsl preconditions written trustinsoft verification kit typedef struct int unsigned long unsigned long buf number rounds aes round keys unaligned data int requires ctx requires buf requires buf requires requires mode mode mode requires length length requires length requires requires requires input length requires input length requires output length ctx int mode length unsigned char const unsigned char input unsigned char output fig acsl preconditions function aes crypt cbc specification function aes crypt cbc provides encryption decryption buffer according aes cryptographic standard cbc encryption mode function takes six parameters last two input output strings parameter ctx stores necessary information aes network particular number rounds round keys defined dedicated structure lines parameter mode indicates whether function encrypt decrypt input parameter length indicates length input string finally parameter provides initialization vector output characters unsigned char declared length actually meaningless tools array typed parameter adjusted pointer type section also footnote page cfp nevertheless considers part specification order generate precise context acsl annotations enclosed special kind comments therefore ignored compiler function precondition introduced keyword requires right function declaration definition must satisfied every call site given function function aes crypt cbc precondition clauses whole function precondition conjunction clauses may tagged names logically meaningless provide way easily refer document specifications instance first precondition line named ctx valid second line named ctx init detail meaning precondition clause pointers must valid properly allocated point memory block appropriate length program safely access either mode predicate mode predicate purpose preconditions ctx valid valid input valid output valid ctx must point memory block containing least single aes context struct must able contain least unsigned characters ranging input output must able contain least length unsigned characters ranging length memory locations read function must properly initialized purpose precondition clauses ctx init init input init initialize first cells buf well every valid cell input specification clause mode specifies mode must either encryption decryption specification clause length mod specifies length multiple block size specified clauses restrict perimeter analysis order make tractable clause ctx standard equality aes context clause ctx true encryption keys finally clause length aims restrict analysis buffers size unsigned characters context generation naive approach context generation would consider one precondition clause directly implement code however would work general since requirements treated order running example instance variables input output depends variable length precondition clauses latter must treated former well generated code variables must initialize latter first former afterwards sound solve problems one could first record every dependency among involved specification proceed generate code accordingly approach based dependency graph nonetheless insufficient preconditions need inference reasoning order implemented correctly example treating precondition requires demands infer array elements order consider initialization correct give overview treat context generation means cfp aes crypt cbc function contract cfp provides result shown figure assuming size unsigned long first note every execution path ends call function aes crypt cbc calls code initializes context variables prefixed cfp order satisfy precondition function different paths contribute cover cases specification initialization code generated sets constraints first inferred every involved precondition inferring constraints precondition clauses implicit dependencies among made explicit recorded dependency graph latter finally visited guide code generation process order obtain correct code let start detailing generated code preconditions length figure lines first cfp declares variable cfp length type length line initializes means library function frama unsigned int interval line takes two unsigned int arguments returns random value comprised two allows fulfill kind information customizable within int void unsigned char unsigned char int char char unsigned char malloc unsigned char char unsigned char malloc int else int return fig slightly simplified version code generated cfp specification figure compared actual version integer casts removed reasons brevity former requirement guarantee abstract interpreters interpret result exactly required interval also corresponds way expert engineers would write general context analyzers finally requirement length implemented conditional line lines implement preconditions ctx pointer aes context instead allocating pointer generated code declares local variable cfp ctx passes address function calls automatically satisfies precondition pointer validity line initializes first bytes structure field buf using library function frama make unknown assuming size unsigned long bytes bytes size values type unsigned long expert engineer would also use library function lines initialize fields single assignments cfp fulfills equality requirement buf respect instead buf latter already refers memory buffer requirements function arguments input output implemented lines let point cfp defines respective variables ctx array unsigned char ctx input ctx output pointers dynamically allocated memory buffers indeed cfp infer exact dimension former specification dimension latter depends value ctx length determined runtime last part generated code lines handles requirement mode either although generated conditional may seem excessive case particular values nonetheless required general case instance consider formula mode mode simplifying acsl preconditions state constraints section presents way systematically reduce function precondition set constraints function context function parameters global variables first introduce specification language shall formalize solution define notion state constraint form requirement turn generate code initializing order simplify state constraints make use symbolic ranges originally introduced blume eigenmann compiler optimization finally provide system inference rules formalizes simplification process core specification language work shall consider specification language figure almost subset acsl predicate defined subsumes acsl predicates see predicates cop defined term comparison cop defined logic formula terms bop integer constant memory value arithmetic operation bop memory values single displacement displacement range variable dereference types integer pointer fig predicates terms types predicates logic defined top typed term comparisons predicates defined terms arithmetic expressions combining integer constants memory values means classic arithmetic operators memory values include variables pointer dereferences memory displacements operator particular defines set memory values may appear outermost construct predicate defined integers defined holds whenever initialized pointers defined holds whenever properly allocated initialized memory region vii term typing terms language typed may take either integer pointer type memory values pointers omit typing rules terms quite standard let specify memory values form pointer type well recursive occurrence must integer type memory values typed set pointers since consider kind coercion construct terms pointer type appear integer terms expected appear arithmetic expressions also follows term comparisons relate terms type term normal forms sake concision simplicity remainder work assumes simplifications take place terms order consider term normal forms particular arithmetic expressions maximally flattened factorized means constant folding techniques conveniently write single displacements also assume memory values displacement ranges either form end terms form simplify finally memory values normalize disjunctive normal forms precondition conjunction predicate clauses one given acsl requires example figure preliminary step shall rewrite conjunctive clause disjunctive normal form pij pij predicate literal simply literal predicate without nested logic negative literal either form pointers every negative literal input predicates translated positive literal applying standard arithmetic logical laws literal called positive literal rest section focuses positive literals negative literals conjunctive clauses handled end disjunctive clauses considered discussing code generation section state constraints interested simplifying predicate literal set constraints leftvalues called state constraints meant indicate minimal requirements resulting function context must implement satisfying function precondition section turn converted code intuitively consider state constraint represent domain definition resulting function context state since domains might determined terms integer constants shall found definition notion symbolic ranges want simplify state constraints define terms symbolic range algebra proposed definitions nonetheless significantly different even though inspired work symbolic expressions symbolic expression defined following grammar bop max min respectively largest smallest expression operators denote set symbolic expressions bop max min viii rest section assume mapping memory values respective symbolic expression let context discriminate former latter section shall simplify symbolic expressions need domain structure let denote define valuation symbolic expression every map obtained substituting every variable distinct integer symbol natural number strictly greater multiplicative coefficient interpreting operators bop min max respective functions denote standard ordering relation preorder defined follows partial order therefore one induced merging equivalence class elements example elements min equivalent lattice symbolic expression ranges symbolic range pair symbolic expressions denoted otherwise said symbolic range interval guarantee denote set symbolic ranges extended empty range partial ordering usual partial order possibly empty ranges symbolic range therefore equivalent consequently domain infimum supremum denote join meet operators respectively worth noting given four symbolic expressions following equations hold min max max min words min max compliant ordering relations section simplifying literals introduced soon incomparable associated resulting unsimplifiable constraint also worth noting general statically computable operators solve practical issue computable symbolic expressions cfp relies equations order delay evaluations runtime eventually code generator convert conditionals state constraints symbolic ranges runtime checks symbolic ranges capture minimal requirements function precondition integer typed symbolic range represents integer variation domain pointer typed represents region valid offsets commonly used abstract interpreters range region analysis respectively however predicate literals simplified symbolic ranges requiring encoding runtime checks verified runtime means conditionals denote rtc cop runtime check two terms call state constraint pair given symbolic range set runtime checks denote resp first resp second projection resp inferring state constraints formalize solution simplifying positive literal set state constraints system inference rules negative literals well conjunctive clauses handled separately end section simplification judgments simplification rules given judgments form predicate literal maps state constraints judgment associates set state constraints literal result simplifying respect appearing updated map equal state constraints latter figures shows formalization main literal simplifications system assume consistency precondition inconsistent rule applies simplification process fails predicates defined figure provides simplification rules literal defined rules variable ereference enforce initialization terms symbolic range neutral ival latter respectively defined pointer type integer type quite common initial approximations inferring variation domains either memory integer values rules ange ange enforce validity memory region determined displacement range first premise rules established whether already enforced alias memory value indicated singleton range rule ange first enforces initialization soundness displacement bound determined updates region valid offsets pointed include range practice predicates added statically provable moreover note consider lower bound symbolic range memory regions must start index rule ange handles case alias enforcing validity memory region determined take account displacement range particular since single displacements may appear memory equality predicates rule form validity alias within range obtained requiring validity displacement range min max rule dempotence provided allow inference process progress term comparison predicates rules figure formalize simplification integer term comparison memory equality predicates first two actually rule schema describe term comparison simplifications integer comparison operators strict operators treated terms nonstrict ones let detail rule respect generic operator cop rule applies whenever cop rewritten means classic integer arithmetic transformations cop relation cop integer term reduces symbolic range respect one given ival cop latter function takes comparison operator cop integer dempotence variable type neutral ival defined defined ereference defined type neutral ival defined ange defined defined ange base offset defined min max defined simplification literal defined cop defined defined cop ival cop cop defined rtc cop cop base offset defined defined simplification term comparison memory equality literals efined defined defined defined base simplification negative literals fig simplification literals state constraints term arguments returns result symbolic range cop resp cop resp since integer typed terms aliasing issue rule always applied although normally consider case rule conservatively enforces validity term comparison means runtime check aliasing rule handles aliasing two pointers single displacement assuming form distinct pointer first defined single displacement latter determined summing offsets together pointer enforced defined case actual region pointed established larger one pointed considered alias although rather conservative due fact statically computable general second last premise important ensuring soundness negative literals figure shows rules negative literals rules simplify literals state constraints rather ensure precondition consistency instance inconsistent defined value undefined time case system must prevent code generation rule efined checks memory value appear map suffices ensure yet defined rule applies hypothesis pointers determine different memory regions particular two aliases whenever base address one pointer overlap memory region conjunctive clauses either positive negative literals handled sequentially following rule given definition efined assumes negative literals treated positive ones exhaustively applying rule first rule efined afterwards dependency graph memory values conjunctive clause system inference rules figure generates map also computes dependency graph memory values considering formalization section memory values graph actually however considering separately acsl predicates instead defined true anymore graph necessary ensuring first soundness rule system respect mutual dependency consequently correct ordering initializations generating code section generally speaking time rule needs inference used state constraint derivation ereference ange etc edges every involved premise added dependency graph derivation fails soon latter operation makes graph cyclic xii example applying inference system example figure final map associates integer length rtc length array input length along dependency graph figure ctx ctx input length input length buf mode input length output output length buf fig dependency graph aes crypt cbc preconditions generated cfp system inference rule figure sound given conjunctive clause simplification procedure always terminates either fails former case state constraints satisfy respective literals denote theorem conjunctive clause either fails generating code state constraints section presents general scheme implementing preconditions state constraints language enriched one primitive function handling ranges practice primitive meant characterize state constraints precisely possible example report case tool cfp however sake conciseness neither detail formalize code generation scheme nevertheless believe provided explanation enough understand implement system similar setting generating code conjunctive clause consider conjunctive clause pair respectively given map state constraints dependency graph inferred system rules figure shall show general case disjunctive normal forms later generate semantically correct code topologically iterate leftvalues follow dependency ordering every visited consider associated state constraint symbolic range handled generating statements initialize constructs statements actually single assignment although loop assignment may sometimes needed initializing range array cells particular initializations symbolic ranges implemented means primitive function make range integer pointer type practice function must provided analyzer context generated executed symbolically analyzer abstract state associate abstract xiii values respective finally conditionals generated initialize symbolic expressions involving min max initialized rest code guarded conditionals generated runtime checks resume generation scheme following initialization assignments runtime checks code initializing next initialization last function consideration running example function aes crypt cbc called required arguments handling disjunctions rewrite preconditions disjunctive normal form preliminary step process disjunct independently applying inference system figure code generation scheme previously described describe code generation scheme precondition given code fragments every disjunct code fragment directly generated otherwise additional variable cfp disjunction generated initialized interval switch construct conditional generated case contains fragment respective resume context generated function including following code pattern switch case break case break case break primitives cfp tool cfp follows generation scheme described implements make range terms frama interval integral type frama make unknown handle symbolic ranges integers pointers respectively properly supported two abstract value analysis tools eva implementation evaluation implemented context generation mechanism called cfp context preconditions written approximately lines ocaml although open source cfp due current contractual obligations cfp successfully used company trustinsoft verification kit library open source implementation protocol evaluate approach particular cfp terms quite natural properties usefulness efficiency quality generated contexts work provides first formal answer practical recurring problem analyzing single functions indeed acsl subset considered expressive enough programs importantly cfp enables tool support compelling fragment acsl minor expense implementing two particularly compared implementation native support ever possible finally cfp proved useful operational industrial setting revealing mistakes contexts previously written hand expert verification engineers xiv although disclose precise data latter cfp revealed notably overlooked cases disjunctions led fix incomplete specifications cfp able efficiently handle rather complex acsl preconditions generation contexts one figure usually instantaneous although disjunctive normal form exponentially larger original precondition formula transformation used practice leads better code terms readability tractability verification tools approach justified fact practice small number disjuncts typically used acsl specifications approach allows generate contexts reasonably readable follows code patterns experts framework use manually write particular handling disjunctions cfp factorizes generated code particular soon rule system infers solution conjunctive clause instance running example initialization variable mode depends disjunction mode mode hence initialized considering cfp disjunction figure conclude briefly discussing current limitations acsl fragment considers quantifier free predicate coercion constructs allowed support casts among integer easy add whereas treating memory addresses integers notoriously difficult leave future work related work similarly approach program synthesis automatically provides program fragments formal specifications however two approaches different purposes executed either symbolically concretely synthesized program provides one computational state satisfies specification context must characterize states particular every state must satisfy specification conversely set states must contain every possible one software testing contexts useful concentrating testing effort particular inputs test input generation tools like cute pathcrawler allow express contexts functions however user must manually write others like pex directly compile formal preconditions runtime checking tool stady shares elements approach instruments functions additional code ensuring postconditions compliance allowing monitoring test generation however tool performs simple translation neither take account dependencies among inferences domain definition conclusion paper presented novel technique automatically generate analysis context formal precondition function core system formalized provide enough details code generation allow similar systems implemented future work includes formalization code generation well statements proofs fundamental properties system whole running example real world also illustrated presentation whole system implemented cfp generates code close possible code used operational industrial setting already revealed mistakes contexts previously written hand expert verification engineers acknowledgments part research work leading results received funding project french dge bpifrance authors thank trustinsoft support particular pascal cuoq benjamin monate anne pacalet providing initial specification test cases insightful comments thanks anonymous reviewers many useful suggestions advice references barnett halleux logozzo tillmann exploiting synergy icse baudin monate moy prevosto acsl specification language http blazy yakobowski structuring abstract interpreters state value abstractions vmcai blume eigenmann symbolic range propagation ipps botella delahaye kosmatov mouy roger williams automating structural testing programs experience pathcrawler ast canet cuoq monate value analysis programs scam cousot cousot abstract interpretation unified lattice model static analysis programs construction approximation fixpoints popl cuoq result graphs abstract static analyzer jfla delahaye kosmatov late treatment precondition dynamic symbolic execution cstva iso ansi standard technical report http kirchner kosmatov prevosto signoles yakobowski software analysis perspective formal aspects computing kuncak mayer piskac suter complete functional synthesis pldi logozzo pentagons weakly relational abstract domain efficient validation array accesses sac maffra santos barbosa gonnord pereira validation memory accesses symbolic analyses sigplan petiot botella julliand kosmatov signoles instrumentation annotated programs test generation scam polikarpova kuraj program synthesis polymorphic refinement types pldi pugh practical algorithm exact array dependence analysis comm acm rugina rinard symbolic bounds analysis pointers array indices accessed memory regions pldi sen marinov agha cute concolic unit testing engine fse arnold tancau bodik saraswat seshia sketching stencils pldi trustinsoft polarssl verification kit technical report http
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eldan stochastic localization kls hyperplane conjecture improved lower bound expansion dec yin tat santosh december abstract show kls constant isotropic logconcave measures improving current best bound log corollaries obtain improved bound estimate constant exponential concentration constant alternative proof bound isotropic constant also follows ball walk sampling isotropic logconcave density converges steps warm start introduction isoperimetry subset ratio measure boundary subset measure subset complement whichever smaller minimum ratio subsets cheeger constant also called expansion isoperimetric coefficient fundamental constant appears many settings graphs convex bodies plays essential role many lines study geometric setting kls hyperplane conjecture asserts distribution logconcave density minimum expansion approximated halfspace universal constant factor thus conjecture true cheeger constant essentially determined simply examining hyperplane cuts precisely statement use absolute constants norm matrix conjecture logconcave density covariance matrix def inf min isotropic logconcave density eigenvalues covariance matrix equal conjectured isoperimetric ratio absolute constant note isoperimetric constant kls constant reciprocal minimum expansion cheeger constant convenient comparisons constants conjecture formulated kannan simonovits course study convergence random process ball walk convex body proved following weaker bound theorem logconcave density covariance matrix kls constant satisfies isotropic distribution theorem gives bound conjecture says conjecture several important consequences implies ball walk mixes steps warm start isotropic convex body logconcave density best possible bound tight hypercube kls conjecture become central modern asymptotic convex geometry equivalent bound spectral gap isotropic logconcave functions although formulated due algorithmic motivation implies several conjectures asymptotic convex geometry describe next conjecture also known variance hypothesis says following microsoft georgia research university washington yile yintat tech vempala conjecture random point isotropic logconcave density def kxk implies random point isotropic logconcave density lies annulus thin shell constant probability noting kxk var conjecture equivalent asserting var kxk isotropic logconcave density following connection current best bound guedon milman improving line work started klartag eldan shown reverse inequality holds approximately sense namely worst possible kls constant isotropic logconcave densities bounded estimate within roughly logarithmic factor dimension results current best bound log weaker inequality shown earlier bobkov see also slicing conjecture also called hyperplane conjecture following conjecture constant convex body unit volume contains hyperplane section least constant volume equivalently convex body unit volume covariance matrix isotropic constant isotropic constant general isotropic logconcave density covariance multiple identity defined best current bound due klartag improving bourgain bound log study conjecture played influential role development convex geometry past several decades shown ball kls conjecture implies slicing conjecture recently eldan klartag showed thin shell conjecture implies slicing therefore alternative stronger proof kls implies slicing next conjecture bound constant logconcave distributions conjecture constant isotropic logconcave density def varp smooth sup shown maz cheeger constant bounded twice kls constant current best bound kls bound finally conjectured lipschitz functions concentrate isotropic logconcave densities conjecture lipschitz concentration function isotropic logconcave density gromov milman showed also bounded kls constant see lemma background conjectures refer reader results prove following bound conjectured form theorem logconcave density covariance matrix isotropic gives bound improving current best bound following corollary immediate corollary logconcave density isotropic slicing constant poincare constant constant concentration coefficient bounded mention algorithmic consequence mixing time ball walk sample isotropic logconcave density warm start approach kls conjecture true gaussian distributions generally distribution whose density function product gaussian density logconcave function known expansion fact used crucially gaussian cooling algorithm computing volume convex body starting standard gaussian restricted convex body gradually making variance gaussian large enough effectively uniform convex body interest overall strategy similar spirit start arbitrary isotropic logconcave density gradually introduce gaussian term density smaller smaller variance isoperimetry resulting distribution sufficient time good since large gaussian factor crucially related isoperimetry initial distribution achieve latter would like maintain measure fixed subset close initial value distribution changes proof uses localization approach proving inequalities particular elegant stochastic version introduced eldan used subsequent papers fix subset original space original logconcave measure measure without loss generality due result standard localization bisect space using hyperplane preserves volume fraction limit process logconcave measures needles inequalities much easier prove approach runs major difficulties proving kls conjecture original measure might isotropic measures could principle variances roughly equal trace original covariance long thin needles much weaker inequalities hold stochastic localization viewed continuous time version process step pick random direction multiply current density linear function along chosen direction time distribution viewed spherical gaussian times logconcave function gaussian gradually reducing variance gaussian becomes sufficiently small variance overall distribution good isoperimetric coefficient determined inverse gaussian standard deviation inequality shown using standard localization important property infinitesimal change step balance density time martingale therefore expected measure subset original measure time measure set random quantity deviates original value time main question direction use step measure remains bounded gaussian part density small variance show simplest choice namely pure random direction chosen uniform distribution suffices analysis needs potential function grows slowly still maintains good control spectral norm current covariance matrix direct choice kat covariance matrix distribution time hard control use gives improved bound appendix show third moment assumption implies improvement via localization preliminaries section review basic definitions theorems use stochastic calculus paper consider stochastic processes given stochastic differential equations given stochastic processes quadratic variations stochastic processes defined lim lim stochastic partition real numbers maxn called mesh limit defined using convergence probability note first arxiv version paper incorrectly claimed assumption lemma defined via polarization example processes satisfy sdes dxt dwt dyt dwt wiener process sde dxt dwt dyt dwt lemma formula let semimartingale twice continuously differentiable function dxi dxi dxi dxj next two lemmas facts wiener processes first reflection principle lemma reflection principle given wiener process sup second decomposition lemma continuous martingales theorem dambis theorem every continuous local martingale form wiener process logconcave functions lemma dinghas leindler convolution two logconcave functions also logconcave particular linear transformation marginal logconcave density logconcave next lemma logconcave densities folklore see lemma logconcave moments given logconcave density positive integer kxk kxk following elementary concentration lemma also version lemma logconcave concentration isotropic logconcave density kxk much stronger concentration bound shown paouris lemma isotropic logconcave distribution kxk following inequality bounding small ball probability lemma thm isotropic logconcave density kxk absolute constants definition define supremum kls constant isotropic logconcave distributions next lemma follows fact constant bounded kls constant lemma lemma matrix isotropic logconcave density kaxk rank prove lower bound expansion suffices consider subsets measure follows concavity isoperimetric profile quote theorem thm applies even generally riemannian manifolds suitable assumptions theorem cheeger constant logconcave density achieved subset measure matrix inequalities symmetric matrix define namely matrix formed taking absolute value eigenvalues matrix define span rows null space vector positive matrix lemma matrix inequality given symmetric matrices lemma inequality given positive matrices since following lemma stated differently show proof eldan completeness lemma given symmetric matrix positive matrix proof without loss generality assume diagonal hence ajj bij ajj bij aii bij ajj bij eldan stochastic localization section consider stochastic localization scheme introduced slightly general terms discrete localization idea would restrict distribution random halfspace repeat process stochastic localization discrete step replaced infinitesimal steps renormalization linear function random direction one might view informally averaging infinitesimal needles discrete time equivalent would sufficiently small random gaussian vector using approximation see time process introduces negative quadratic factor exponent gaussian factor time tends distribution tends concentrated gaussian eventually delta function point subset measure either idea proof stop time large enough strong gaussian factor density small enough ensure measure set changed constant process basic properties given distribution logconcave density start time distribution time apply infinitesimal change density done picking random direction gaussian certain covariance matrix called control matrix section use process section use varying get bound distributions order construct stochastic process assume support contained ball radius exponentially small probability outside ball lemma moreover since theorem need consider subsets measure truncation affect kls constant distribution definition given logconcave distribution define following stochastic differential equation dct dwt dbt probability distribution mean covariance defined ect kxkbt ect kykbt control matrices symmetric matrices specified later section consider process case ect ect also since bounded function lipschitz respect hence standard existence uniqueness theorems sec show existence uniqueness solution time general following result lemma existence uniqueness compact support bounded lipschitz functions stochastic differential equation unique solution defer proofs statements section considered standard stochastic calculus section proceed analyzing process parameters evolve roughly speaking first lemma says stochastic process continuously multiplying random infinitesimally small linear function lemma lem dpt dwt considering derivative log see applying dpt lemma results distribution gaussian term density log dpt dwt dwt xdt dwt dct dbt xdt last term independent first two terms explain form appearance gaussian next analyze change covariance matrix lemma dwt dat bounding expansion plan bound expansion spectral norm covariance matrix time first bound measure set initial measure lemma set proof let dwt dgt dwt integral might subset hence max max dxdt max hence dambis theorem exists wiener process distribution using used reflection principle brownian motion concentration normal distribution namely theorem let standard gaussian density let logconcave function define density function follows fix unit vector let using derive following isoperimetric inequality proved also used theorem thm let integrable logconcave function positive definite logconcave measurable subset min words expansion proof proof uses localization lemma reduce statement statement gaussian times logconcave density gaussian projection gaussian logconcave function might different limit localization original function along interval times exponential function apply inequality one dimension theorem prove resulting distribution variance gaussian therefore isoperimetric constant bounded inverse standard deviation times constant complete proof general terms carried thm prove bound expansion lemma given logconcave distribution let defined definition using initial distribution suppose proof milman theorem suffices consider subsets measure consider measurable subset initial measure lemma martingale therefore next definition theorem shows expansion hence min false false lem used assumption end using theorem shows controlling via potential section use control matrix third moment bounds two key lemmas tensor distribution special case first inequality used main theorem first lemma need general case proof section lemma given logconcave distribution mean covariance symmetric matrix proof first consider case taking follows isotropic log concave distribution statement becomes calculate max max used fact fixed logconcave distribution lemma hence lemma shows general symmetric matrix write hence vit eigenvalues eigenvectors vit vit lemma given logconcave distribution mean covariance proof without loss generality assume fixed random follows logconcave distribution lemma hence lemma shows next note follows logconcave distribution lemma hence lemma shows analysis using formula lemma one compute derivatives since similar calculation appears sections prove common generalization lemma lemma let defined definition dwt lemma given logconcave distribution covariance matrix let defined definition using initial distribution universal constant max proof let lemma dwt def vtt dwt drift term lemma shows universal constant note dropped term since positive semidefinite therefore term negative martingale term vtt dwt note kvt kat lem drift term grows roughly stochastic thus bounds drift grows term term stochastic term suggest potential remains formalize decoupling two terms let formula kvt vtt dwt dyt dyt universal constant note kvt theorem exists wiener process distribution using reflection principle brownian motion max max exp since kap therefore shows max exp putting max exp note implies hence max proof theorem proof theorem rescaling assume lemma max since implies kat min kas theorem follows lemma localization proofs begin proof existence unique solution sde proof lemma write stochastic differential equation dct dwt dbt kykb since compact support lipschitz variables functions next note bounded since compact support since bounded lipschitz function variables therefore use standard existence uniqueness theorem sec show existence uniqueness solution time next proof infinitesimal change density proof lemma let ect kxkbt formula applied ctt hbt dqt dctt hdbt def note hence quadratic variations ctt dctt dwt ctt hct also dbt predictable process namely stochastic term hence hbt therefore gives dqt dwt let dvt dqt dwt dwt formula dvt dwt hct dwt combining dpt dqt dwt dwt dwt next proof change covariance matrix proof lemma recall viewing function variables apply formula derivation use denote matrix whose coordinate similarly column vector row vector dpt dat factor comes hessians second term vanishes similarly third term also vanishes compute last terms note xpt dwt dwt dwt dwt therefore last term att simply write similarly gives fourth term similarly fifth term dtdx combining terms dpt dat next proof stochastic derivative potential lemma let defined definition integer dtr dwt qtr proof let first directional derivatives given qtr using formula dtr qtr dat ijkl eij ekl aij akl eij matrix entry otherwise aij stochastic process defined entry using lemma lemma dat dwt dwt coordinate therefore aij akl using formula dat aij akl dtr dwt qtr eij ekl ijkl dwt qtr acknowledgement thank ravi kannan laci assaf naor nisheeth vishnoi continuous support encouragement also thank bubeck ben cousins ronen eldan klartag anup rao leonard wong helpful discussions references zeyuan yin tat lee lorenzo orecchia using optimization obtain parallel simpler faster positive sdp solver proceedings annual symposium discrete algorithms pages siam david bastero approaching variance conjectures volume springer shiri apostolos giannopoulos vitali milman asymptotic geometric analysis part volume ball logarithmically concave functions sections convex sets studia mathematica bobkov isoperimetric constants probability distributions geometric aspects functional analysis lect notes bobkov koldobsky central limit property convex bodies pages springer berlin heidelberg berlin heidelberg bourgain distribution polynomials high dimensional convex sets jean bourgain high dimensional maximal functions associated convex bodies american journal mathematics brascamp lieb extensions theorems including inequalities functions application diffusion equationslogarithmic concave measures functions functional silouanos brazitikos apostolos giannopoulos petros valettas vritsiou geometry isotropic convex bodies volume american mathematical society providence jeff cheeger lower bound smallest eigenvalue laplacian pages princeton univ press cousins vempala cubic algorithm computing gaussian volume soda pages cousins vempala bypassing kls gaussian cooling volume algorithm stoc pages eldan thin shell implies spectral gap polylog via stochastic localization scheme geometric functional analysis eldan klartag approximately gaussian marginals hyperplane conjecture contermporary mathematics ronen eldan james lee talagrand convolution conjecture gaussian space ieee annual symposium foundations computer science focs berkeley usa october pages ronen eldan joseph lehec bounding norm vector via estimates fleury concentration thin euclidean shell measures funct gromov milman topological application isoperimetric inequality amer olivier guedon emanuel milman interpolating sharp estimates isotropic measures geometric functional analysis kannan simonovits isoperimetric problems convex bodies localization lemma discrete computational geometry klartag convex perturbations bounded isotropic constant geom funct klartag central limit theorem convex sets invent klartag estimates central limit theorem convex sets funct ledoux spectral gap logarithmic sobolev constant geometric bounds surveys diff vol pages int press lieb thirring inequalities moments eigenvalues equation relation sobolev inequalities studies mathematical physics essays honor valentine bargman lieb simon wightman eds pages simonovits random walks convex body improved volume algorithm random structures volume pages vempala geometry logconcave functions sampling algorithms random structures algorithms irini perissinaki milla anttila keith ball central limit problem convex bodies transactions american mathematical society milman role convexity isoperimetry spectral gap concentration invent bernt oksendal stochastic differential equations introduction applications springer science business media paouris concentration mass convex bodies geometric functional analysis vempala algorithmic aspects convexity lecture notes institut henri poincare winter school january classes domains imbedding theorems function spaces dokl acad nauk sssr engl transl soviet math reduction third moment assumption section use following assumption first arxiv version paper claimed assumption lemma might true proof correct assumption third moment isotropic logconcave distribution note assumption prove following theorem third moment assumption isotropic logconcave density kls log log log constant proof use process sensitive potential function even integers tensor bounds definition isotropic logconcave distribution symmetric matrices define often drop subscript indicate worst case bound def sup isotropic logconcave remark clear definition symmetric namely permutation first start simple equalities repeatedly use elementary facts ayxt lemma isotropic logconcave distribution symmetric matrices aij xxt proof direct calculation shows ayxt byxt ayxt byxi axx byy ayxt byxt aij byxt aij aij byy lemma symmetric matrices proof fix isotropic logconcave distribution define xxt well defined since yxt yxt since symmetric therefore second part write define similarly note since first part lemma shows every term hence lemma suppose given isotropic logconcave distribution unit vector define xxt orthogonal projection matrix rank symmetric matrix log proof first bound part proof generalized proof eldan note since ext lem var gives bound since assume without loss generality write eigenvalues eigenvalues smaller equals clearly need many let orthogonal projection span range using kai rank kai kai used first part lemma last inequality similarly combining bounds rank kai rank kai log log next lemma collect tensor related inequalities useful lemma suppose distribution symmetric matrices isotropic logconcave tra min log proof without loss generality assume diagonal rotating space particular want prove something symmetric matrices use spectral decomposition rewrite puts back situation diagonal matrix let xxt inequality note aii ajj lem inequality note tra hence lem tra var remaining inequalities suffices upper bound upper bounding isotropic logconcave distribution inequality note lem aii lem last inequality third moment assumption inequality note lem aii lem inequality let orthogonal projection span range let rank lem lem lem used inequality note lem aii log lem inequality note lem lem lem lemma positive matrices lem lem proof fix isotropic logconcave distribution let xxt yxt yxt xxt using lemma yxt byxt taking supremum isotropic logconcave distributions get result derivatives potential lemma let defined definition integer dtr dwt qtr give proof section next lemma bounds stochastic process controls potential function lemma let defined definition let even integer vtt dwt kvt proof lemma dwt qtr dwt qtr yxt yxt ydt def vtt dwt isotropic version defined drift term vtt dwt martingale term drift term qtr first term drift lem lem second term drift since even qtr qtr qtr martingale term vtt dwt note kvt analysis first bound drift term lemma lemma suppose let even integer log proof lem used lemma end first term log log lem lem log used last line second term write consists eigenvalues remaining part pick later first term note lem lem second term note used lem lem lem last line third term similarly lem lem lem fourth term let orthogonal projection range lem lem using combining four terms balancing last two terms setting get used third term lem lem set last line fourth term lem lem set last line combining terms result next bound martingale term lemma let logconcave distribution covariance matrix let even integer proof note lem using lemma lemma know satisfies stochastic equation vtt dwt log kvt log used using one bound growth using stochastic inequality completeness bound directly lemma suppose given isotropic logconcave distribution let defined definition using initial distribution let even integer large constant universal constant max log proof idea choose function resulting stochastic equation effectively decouples drift martingale terms use etq formula dwt used concave last line dropped second derivative term rationale choice etq guess solution sde power chosen eliminated stochastic term bound use get term etq etq etq log first term note etq etq used end second term note etq etq used end third term assuming etq etq etq etq used implies fourth term assuming log etq log log etq etq used end combining four terms term etq term dwt using assuming kvt hence combining terms etq dwt dyt martingale universal constant theorem exists wiener process distribution using reflection principle brownian motion max max exp let long estimate valid hence max max used last line note setting log using large constant hence max max exp note log log used using fact max exp used large constant lemma suppose even integer large constant log universal constant proof lemma log probability assuming event kat less get kat log large enough also hence apply lemma log since argument holds isotropic logconcave distribution gives bound proof theorem fix large enough start known bound universal constant larger hence apply lemma every log repeating process induction log log hence log setting log log log log log log log log log exp adaptive localization anisotropic distributions section show third moment assumption gives following bound kls constant arbitrary logconcave distributions theorem third moment assumption section logconcave density covariance matrix integer kls constant bounded controlled stochastic localization definition given symmetric matrix let span eigenvectors eigenvalues less dim define etc similarly reduction apply localization subspace matrix controlling gaussian small eigenvalues time control matrix chosen inverse projection current covariance matrix subspace small eigenvalue captured next definition definition given logconcave distribution threshold define following process inf defined definition initial data instead control matrix given lim covariance matrix orthogonal projection onto known bound kls constant isotropic logconcave densities let rank following lemma gives alternative definition lemma orthogonal projection matrix let denotes pseudoinverse furthermore rowspace equals nullspace proof taking see lim lim hence suffices prove case diagonal matrix whose first diagonal entries remaining diagonal entries write matrix hence lim first coordinates hence kxk taking using shows another hand hence taking limit shows hence specific formula important reduction section uses following properties control matrix lemma focus small values focus small values large step size trct proof first part since second part prove continuous induction let inf kxk suppose definition fix since write eigenvalues since since number eigenvalues unchanged know definition hence lemma shows hence since dbt kxk contradicts definition therefore second part lemma shows therefore dim rank used number eigenvalues unchanged end third part use inequality xqi fact rank matrix trct use denote power pseudo inverse lemma hence lem putting gives trct analysis lemma let defined definition integer dtr aqt dwt qtr proof note defined concatenating solutions finitely many sdes therefore suffices prove equality sde solution follows lemma section may positive hence need take even integer section analyze process potential aqt hence require even lemma let defined definition let aqt integer vtt dwt kvt proof lemma dwt qtr dwt qtr ydt vtt dwt def isotropic version defined drift term vtt dwt martingale term drift term using qtr aqt lem aqt lem lem martingale term vtt dwt note kvt lem used last line using one bound growth lemma let defined definition using initial distribution let aqt integer suppose universal constant max tmax tmax proof lemma formula log vtt dwt dwt dyt dyt dwt universal constant note kvt universal constant theorem exists wiener process distribution using reflection principle brownian motion max max exp since log log therefore shows max log log exp putting max log log exp proof theorem lemma let defined definition using initial distribution suppose aqt btmax tmax universal constants proof lemma lemma tmax probability subject event trct let trbt shows theorem log log log universal constant hence log log log also lemma shows therefore let inf since log log log increasing function log log log log log log definition see log log log log log log therefore log log log last inequality seen noting sequence exponentially increasing rate maximal setting tmax hence increases time tmax increased localization process freezes proof theorem case proven theorem assume rescaling assume traq apply lemma note lemma btmax tmax furthermore lemma shows therefore tmax btmax hence lemma shows traq
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image registration techniques survey sayan nag department electrical engineering jadavpur university image registration process aligning two images scene reference particular image images captured various sensors different times multiple thus get better picture change considerable period time image registration important image registration finds application medical sciences remote sensing computer vision paper presents detailed review several approaches classified accordingly along contributions drawbacks main steps image registration procedure also discussed different performance measures presented determine registration quality accuracy scope future research presented well registration classification contribution drawback performance measures registration quality accuracy future research introduction image registration interpreted process overlaying two images scene respect particular reference image images may taken various circumstances various perspectives additionally various sensors reference image generally one captured images geometrically transforms different sets data particular reference system discrepancies among images interposed owing disparate imaging conditions image acquisition devices underwent rapid modifications proliferating amount diversity acquired images elicited research automatic image registration image analysis ventures one significant step image registration necessary step obtain final information combination multitude divergent sources capturing information varied circumstances diverse manners essentially objective detect concealed relationship existing input reference images usually indicated coordinate transformation matrix accordingly image registration essentially devised optimization problem image registration plays crucial role many applications image registration finds applications remote sensing involving multispectral classification environmental corresponding author mail monitoring change detection image mosaicing weather forecasting creating images integrating information geographic information systems gis medicine including fusion computer tomography nmr data obtain complete information patient analysis different diseases like epilepsy protocols incorporate functional data along anatomical mri monitoring tumor evolution treatment verification juxtaposition patient data anatomical atlases cartography map updating computer vision target localization automatic quality control motion tracking according manner image acquisition application image registration segregated following groups analysis images similar object scene captured multiple viewpoints gain better representation scanned object scene examples include mosaicing images shape recovery stereo analysis images scene captured various times usually dissimilar conditions notice changes scene emerged successive images acquisitions examples include motion tracking tracking growth tumors analysis different sensors used acquire images merge information obtained various sources obtain minutiae examples include integration information sensors disparate characteristics providing better spatial spectral resolutions independent depends upon robustness registration algorithm combination sensors capturing anatomical information like magnetic resonance image mri ultrasound sensors acquiring functional information like positron emission tomography pet single photon emission computed tomography spect magnetic resonance spectroscopy mrs study analyze seizure disorders alzheimer disease depression diseases figure shows example registration section presents steps involved image registration section contains classification criteria registration methods presented section transform model estimation performance analysis discussed sections respectively section contains conclusion fig multimodal dots represent anatomical landmarks fiducial points axial view brain image anatomical information pink dots represent meg sensors locations green dots represent sensors locations meg eeg data contain functional information bottom picture shows coregistered brain image sagittal view steps involved image registration image registration task involves following steps follows feature detection important task image registration process detection process manual automatic depending upon complexity though automatic detection features preferred regions edges contours line intersections corners along point representatives like center gravity line endings collectively known control points serve features features consisting distinctive objects must easily detectable features physically interpretable identifiable feature set reference image must sharing sufficient common features image irrespective undesired occlusions unexpected changes proper registration algorithm detection robust enough able detect features projections scene without affected specific image deformation degradation feature matching step essentially establishes correspondence features detected sensed image detected reference image different feature descriptors similarity measures besides spatial relationships among features adopted set accurate accordance feature descriptors must formulated remain unchanged spite degradations concurrently must able properly discriminate among diverse features remaining unaffected noise transform model assessment alignment sensed image reference image parameters mapping functions estimated parameters computed established feature correspondence obtained previous step selectivity mapping function depends priori knowledge regarding acquisition process expected image deformations absence priori information flexibility model must ensured tackle image deformations image transformation sensed image transformed alignment employing mapping functions mentioned image registration steps generally followed figure shows pictorial representation steps involved image registration though noteworthy mention difficult fabricate universal method applicable registration assignments reason attributed diversity images registered obtained miscellany sources several types degradations introduced images besides geometric deformation images radiometric deformations noise corruptions taken account proper registration images fig steps involved image registration fig steps involved image registration top reference image top image aligned image yellow dots represent extracted features enough common features images mapping function established gives bottom image final output iii classification criteria image registration techniques image registration techniques classified based criteria follows dimensionality specifies dimensions different possible registrations may based requirement domain transformation may global entire image registered may local portion image taken consideration registration purpose type transformation may rigid translation rotation reflection affine translation rotation scaling reflection shearing projective registration quality depending data features extracted several measures adopted applied parameters registration obtained employing search oriented methods optimum parameters found search method heuristic search method determines quality transformation hence registration subject registration subject considered registration subjects different known registration object registration different objects include head abdomen thorax knee etc nature registration basis may extrinsic based foreign objects easily detectable markers glued skin intrinsic based image information based imaging coordinates two devices matched interaction may interactive entirely automatic modalities involved may also termed using modalities like computed tomography magnetic resonance imaging mri positron emission tomography pet single photon emission computed tomography spect ultra sound xray digital subtraction angiography dsa multimodal also known image employing two modalities mentioned methods image registration various methods image registration follows extrinsic methods method artificial foreign objects easily detectable attached patient body serve external features used feature matching complexity lessened hence computational fast accuracy also maintained examples markers glued patient skin frame attached rigidly patient outer skull invasive neurosurgery related purposes surface methods surfaces boundaries contours generally distinct medical images unlike landmarks example approach employed registering multimodality brain image surface matching algorithms generally applied rigid body registration collection points generally called point set extracted contours image two surfaces considered registration two sets surface covering larger volume patient higher resolution volume coverage comparable generally considered generation surface model iterative closest point algorithm correspondence matching algorithm successfully applied registration algorithms techniques evolutionary optimization also seen solve high dimensional optimization problems surface registrations moments principle axes methods orthogonal axes moments inertia minimized known principle axes two identical objects registered accurately bringing principal axes concurrence without employing transformations objects identical similar appearance approximately registered technique moment based methods presegmentation done many cases engender satisfactory outcomes correlation based methods method essentially useful registration monomodal images comparison several images similar object immense usage field medical sciences analyzing treatment disease extracted features images also used obtain coefficients image registration techniques based fourier domain also used image registration successful yet complex ventures significantly made using frequency estimation approach fourier based image registration problem employing multiple signal classification algorithm music proliferate robustness eventually yielding accurate results normalized mutual information images used image registration purposes adopting entropy correlation coefficient ecc techniques accompanied search algorithms exploited evaluate transformation two input images mutual information based methods mutual registration methods joint probability intensities comparable voxels images consideration estimated mutual information based measures utilized aid registration mutual information fruitfully utilized establishing correspondence features reference sensed images mentioned step featurematching correlation methods proved inefficient registration mutual information based methods suffer problem rather found perform effectively registration tasks gradient descent optimization methods employed maximize mutual information window pyramid based approaches used achieve image registration using mutual information methods used include hierarchical search strategies along simulated annealing powell direction set method recently various optimization methods multiresolution strategies adopted mutual information maximization wavelet based methods wavelet transform introduced get idea time instant particular frequency exists width window altered transform computed spectral important characteristic wavelet transform offers time frequency selectivity able localize properties temporal frequency domains image registration effectively choosing several wavelet coefficients selection rules like maximum absolute wavelet coefficient image image individual band partial wavelet coefficients image replaced image pyramidal approaches also use wavelet decomposition owing intrinsic multiresolution properties different types wavelets like haar symlet daubechies coiflets applied finding correspondence different sets wavelet coefficients feature extraction techniques along normalized matching image matching techniques used thereby incorporating sufficient control points reduce local degradations image registration soft computing based methods methods comparatively recent advanced successfully applied image registration tasks include artificial neural networks fuzzy sets several optimization heuristics artificial neural networks artificial neural network ann computational model formulated based biological neural networks also known perceptron mlp since contains number hidden layers layers consist interconnected group artificial neurons information passed one layer next layer artificial neural networks simply neural networks learns adaptively learning phase information flows network updates accordingly assigning various weights neural networks viewed upon statistical data modeling tools employed model complex relationships inputs outputs recognize patterns data also called pattern recognition two types schemes networks links devoid loop multilayer perceptron mlp radial basis function neural networks rbf recurrent networks include loops selforganizing maps som hopfield neural networks priori information output essential requirement training feed forward networks hand recurrent neural networks generally require previous knowledge regarding expected output rigorous training process ann modifies adaptively updates network architecture abreast connection weights link weights able learn complex relationships thereby parlaying robustness efficacy performance multilayer perceptron radial basis functions maps hopfield networks utilized different computational optimization aspects designing registration matrices image registration problems neural networks also used solving monomodal medical image registration problems fuzzy sets fuzzy set collection elements continuous sequence membership grades degrees fuzzy sets introduced zadeh fuzzy sets follow properties inclusion union complement intersection etc classical set theory membership values elements set decided binary terms depending upon whether element belongs belong set contrast fuzzy set theory allows grading membership elements fuzzy set decided assistance membership function assigns values residing interval fuzzy sets manifest perception partial membership element within permits fuzzy sets tackle uncertainty inaccuracies fuzzy sets explicitly applied image registration techniques also utilized choose extracted features registered fuzzy logic used enhance precision transformation parameters estimated approximately previously eventually leading accurate registration estimates optimization heuristics optimization problems applied several domains engineering design optimization mathematical models objective functions may unconstrained without constraints constrained constraints continuous well discrete variables task finding optimal solutions difficult numerous curtailments active points global optima traditional methods including gradient descent dynamic programming newton methods computationally less efficient whereas provide feasible solutions stipulated time list metaheuristics include genetic algorithm particle swarm optimization pso gravitational search algorithm gsa ant colony optimization aco stimulated annealing plant propagation algorithm ppa relatively old approximate search technique used computing global search heuristics form important class evolutionary algorithms mimics evolutionary biological processes mutation selection crossover abandonment likewise particle swarm optimization differential evolution along existing variants relatively advanced heuristics efficiently solve optimization problems optimization heuristics applied image registration problems finding optimal parameters necessary designing transformation model transform model estimation transformation expounded process mapping set points various locations objective design proper transformation model transforms sensed image respect original image maximum accuracy transformations may performed translation rotation scaling shearing reflection collectively known affine transformation also projective transformations well translation let point translated units matrix representation transformation given new point old point translation value rotation point plane rotated angle respect origin relationship final point initial point given new point old point rotational parameter scaling scaling required resize image work images whose voxel sizes differ images represented new point old point scaling parameters shearing shearing parallel lines preserved may represented new point old point shearing parameters fig shows example shearing transformation selection similarity measures depends modality images registered correlation based metrics like correlation coefficient applicable registration mutual information utilized image registration purposes correlation coefficient essentially similarity measure gives idea well reference transformed images identical two images perfectly identical gives value equal whereas two images completely uncorrelated value equal value equal indicates images completely means one image negative gives satisfactory results registration represented intensity ith pixel reference sensed image respectively mean intensity reference sensed image respectively mutual information yet another measure determining degree similarity measured image intensities corresponding voxels images maximized images accurately aligned values symmetric range values starts zero vary high value high value depicts large reduction uncertainty whereas zero value clear indication two variables independent represented joint distribution function marginal distribution functions fig example affine transformations translation rotation scaling shearing performance analysis required estimate accurate registration actually also qualitatively analyze performance algorithms metrics used also serve basis improvement registration iteration fig example image registration using mutual information similarity measure top reference mri brain image axial view top aligned pet brain image axial view transformed pet brain image axial view vii conclusion paper tries present survey registration methods along detailed classifications among various approaches image registration essential step integrating fusing analyzing information various sensors sources immense applications fields medical sciences computer vision remote sensing image registrations complex nonlinear distortions registration registrations occluded images despite affected illumination factors among others thus contributing robustness approaches belong challenging tasks present scenario generation features control points mapping transformation functions essential steps lot research work needs done enhance accuracy multimodal registration technique gained popularity particular whereas images correlation based similarity metrics preferred robustness reliability proliferated hybrid approaches combining based techniques measures several soft computing methods including optimization heuristics applied find optimum parameters mostly case affine transformations based registration gold standard algorithms approaches developed image registration purposes dependency images consideration thus despite lot work done automatic image registration still considered open problem future works introducing new methods apt features provide robust well accurate outcomes registration acknowledgment would like extend sincere gratitude professor sugata munshi professor amitava chatterjee professor mita dutta support guidance references fonseca manjunath registration techniques multisensor remotely sensed imagery photogrammetric engineering remote sensing clerk maxwell treatise electricity magnetism vol oxford clarendon results test image matching isprs isprs journal photogrammetry remote sensing moigne first evaluation automatic image registration methods proceedings international geoscience remote sensing symposium igarss seattle washington hill batchelor holden hawkes medical image registration physics medicine biology lester arridge survey 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unique parallel decomposition matias david lee univ lyon ens lyon cnrs ucb lyon lip france bas luttik eindhoven university technology netherlands fragment process algebra satisfies unique parallel decomposition definable behaviours admit unique decomposition indecomposable parallel components paper prove finite processes processes perform infinite executions satisfy property modulo strong bisimilarity weak bisimilarity results obtained application general technique establishing unique parallel decomposition using decomposition orders introduction fragment process algebra unique parallel decomposition upd definable behaviours admit unique decomposition indecomposable parallel components paper prove finite processes definable satisfy property modulo strong bisimilarity modulo weak bisimilarity theoretical point view property interesting used prove theoretical properties process calculi instance relying unique parallel decomposition moller proves ccs finitely axiomatized without auxiliary operations hirshfeld jerrum prove bisimilarity decidable normed unique parallel decomposition also used define notion normal form notion normal form useful completeness proofs equational axiomatizations settings elimination theorem parallel composition lacking see upd used prove complete axiomatisation decidability results context process calculus practical point view unique parallel decomposition used devise methods finding maximally parallel implementation behaviour improving verification methods unique parallel decomposition result used tool comparison different security notions context electronic voting upd property widely studied different process calculi variants parallel operator milner moller first establish unique parallel decomposition theorem proved property simple process calculus allows specification finite behaviours strong bisimilarity includes parallel composition form pure interleaving without interaction components moller dissertation extended result replacing interleaving parallel composition ccs parallel composition also considering weak bisimilarity christensen also dissertation proved unique decomposition normed behaviours recursively definable modulo strong bisimilarity behaviours recursively definable modulo lee supported project anr pace gebler peters eds combined workshop expressiveness concurrency structural operational semantics eptcs unique parallel decomposition distributed bisimilarity proof latter result relies cancellation law parallel composition distributed bisimilarity first established castellani lemma aforementioned unique parallel decomposition results established subsequent refinements ingenious proof technique attributed milner notion decomposition order introduced order formulate sufficient condition commutative monoids facilitates abstract version milner proof technique proved partial commutative monoid endowed decomposition order unique decomposition thus algebraic tool obtained allows one prove upd process calculus finding decomposition order tool deal settings aforementioned paper show tool also applied obtain unique parallel decomposition results finite processes strong bisimilarity weak bisimilarity end face two complications first complication context opposed previous settings decomposition order directly induced commutative monoid processes transition relation culprit general two parallel components may fuse single indecomposable process result scope extrusion define decomposition order consider fragment transition relation avoids phenomenon second complication arises case weak bisimilarity certain transitions deemed unobservable consequence transitions change state weakly bisimilar processes demonstrate decomposition order nevertheless obtained ignoring stuttering transitions paper studies unique parallel decomposition strong bisimilarity weak bisimilarity applied applied variant designed verification cryptographic protocols main feature channels transmit variables values variables set using active substitutions roughly active substitution extension grammar works memory save value variable variables transition observable memories possible mask sensitive information proof result strong case relies induction norm process fact norm arguments parallel composition less norm parallel composition unfortunately property true restriction operator see section counter example reason restrict finite processes strong setting proof weak case follows proof technique attributed milner general techniques applied directly setting applied due active substitutions second author presented adaptation general result order make suitable establishing unique parallel decomposition settings notion unobservable behaviour ensued technique amounts showing transition relation induces weak decomposition order satisfying property called power cancellation present paper show instead using adapted technique original technique may applied settings notion unobservable behaviour considering fragment transition relation method appears simpler method suggested result paper organized follows section briefly recall abstract framework introduced prove upd results section recall syntax different semantics section composed two subsections section introduce notion depth process prove properties notion section use results result section prove finite processes satisfy unique parallel decomposition strong bisimilarity section follows similar structure section introduce notion processes without stuttering transitions prove properties kind processes properties lee luttik result section used section prove finite processes satisfy unique parallel decomposition weak bisimilarity section present final remarks decomposition orders section briefly review theory unique decomposition commutative monoids shall apply remainder paper prove upd results context definition commutative monoid set distinguished element binary operation denoted associativity commutativity identity remainder paper often suppress symbol use definition element commutative monoid called indecomposable implies definition let commutative monoid decomposition finite indecomposable elements defined element called composition associated decomposition conversely say decomposition element decompositions equivalent notation composition decomposition unique implies decompositions say element unique decomposition decomposition decomposition unique every element unique decomposition say unique decomposition theorem gives sufficient condition ensure commutative monoid unique decomposition requires existence decomposition order definition let commutative monoid partial order decomposition order every subset implies case say element identity element least element respect strictly compatible defined precompositional implies archimedean implies theorem every commutative monoid decomposition order unique decomposition recall syntax rules define transition relation assume set names channels use range unique parallel decomposition inp res open mat tau table transition rules definition processes summations prefixes given respectively denote set processes occurrence name bound process scope restriction input name free process least one occurrence bound write denote respectively set bound names free names process set names process defined employ following convention names convention discussion assume bound names processes actions consideration chosen different names free entities consideration processes actions substitutions sets names convention subject limitation considering transition name bound may occur free limitation necessary expressing scope extrusion transition relation associated term defined rules table omitted symmetric version rules denote set visible actions executed process action internal action define write derivation rules table definition strong bisimilarity largest symmetric relation notation whenever lee luttik relation compatible input prefix whereas received channel hence congruence full syntax however congruence see theorem compatible constructs syntax present paper shall use fact compatible parallel composition recall weak variant bisimilarity write write notice difference second case least one executed definition weak bisimilarity largest symmetric relation notation whenever like strong bisimilarity possible prove congruence contexts see theorem unique decomposition respect strong bisimilarity section shall use result presented section prove every finite process unique parallel decomposition strong bisimilarity section introduce definition depth process properties also explain restrict development finite processes section present unique decomposition result depth process given set denote set finite sequences empty sequence write processes implies interested write addition write definition let length function defined length unique parallel decomposition definition process normed denote set normed processes depth norm normed process defined respectively depth length norm length sup whenever infinite set inf remark assigned higher weight occurrences label definition length sequence ensure depth additive parallel composition depth parallel composition sum depths components shall prove lemma opposed process calculi unique decomposition established see due scope extrusion norm additive consider performs execution infinite normed length ensure kind properties one approach could consider normed processes unfortunately enough consider processes normed moreover perform infinite execution despite norm norm norm moreover notice norm arguments parallel composition less norm parallel composition norm norm particular examples show item lemma norm additive false consequence proofs flawed authors proposed solution problem discuss conclusion paper facilitate inductive reasoning consider finite processes depth definition process finite denote set finite processes following last example depth depth depth conclude section present collection results including lemmas theorems lemmas needed prove theorems theorems lemmas used next section theorem states bisimilar processes depth theorem states depth parallel composition two processes bisimilar greater depth process thanks results able extend notion depth equivalence classes apply inductive reasoning lemma implies depth lemma depth depth theorem iff moreover depth depth lee luttik proof suppose clearly iff hence iff prove depth depth first note depth depth case remains depth depth natural numbers proceed induction depth depth lemma therefore depth suppose depth assume statement holds processes depth less suppose depth length depth definition get lemma depth depth therefore depth depth depth depth length depth get contradiction similarly case depth reach contradiction considering transition length depth conclude depth depth lemma let depth length depth length therefore depth depth length lemma depth depth lemma let length depth length length length proof proceed complete induction depth depth suppose property holds parallel compositions finite processes sum depths smaller let length analyse different ways deriving first transition omit symmetric cases case lemma induction depth length length length length length length length case lemma induction depth length length length length length length length length length length length case side condition allows use rules symmetric version one hand lemma depth depth hand depth depth maximal execution length length xzxz lemma depth depth moreover length depth length point repeat proof first case lemma processes depth depth depth proof lemma ensure depth depth depth hand vention implies allows conclude depth depth depth therefore depth depth depth unique parallel decomposition lemma iff theorem depth depth depth depth proof lemma depth depth theorem depth depth lemma depth depth depth conclude depth depth depth depth unique decomposition commutative monoid associated modulo defined lemma definition sound theorem depth lift function depth depth depth lemma neutral element binary operation commutative monoid satisfies associativity commutativity identity properties order use theorem need define decomposition order shown transition relation directly induces decomposition order commutative monoid processes case however order induced transition relation directly used illustrated following example define binary relation denote inverse reflexivetransitive closure order precompositional consider processes therefore note executes one transition clear processes particularity example scope extrusion need define order based fragment transition relation avoids phenomenon shall define partial order closure relation turn defined follows partial order defined inverse closure write notice definition avoids kind communications arguments parallel operator ensures scope extrusion also avoided lee luttik lemma depth depth lemma partial order proof prove reflexive antisymmetric transitive reflexive transitive closure prove antisymmetric notice implies lemma depth depth therefore implies lemma prove decomposition order prove result need add last auxiliary result lemma lemma depth proof proceed complete induction depth assume hypothesis holds values less suppose given therefore finally define suppose theorem depth depth lemma depth induction therefore definitions therefore define proof complete lemma decomposition order proof prove every subset element let let depth min depth minimal element lemma definition least element consider proceed induction depth depth therefore suppose depth lemma lemma depth depth induction definition strictly compatible suppose consider definition definition define max definition lemma depth depth precompositional suppose prove conditions satisfied suppose one processes bisimilar suppose case define conditions also satisfied suppose definition unique parallel decomposition processes proof proceed induction suppose hypothesis holds prove case given definition either omitted play role suppose induction partial order conclude proof archimedean suppose lemma depth depth given depth conclude depth therefore theorem follows unique decomposition corollary commutative monoid unique decomposition unique parallel decomposition respect weak bisimilarity prove result unique parallel decomposition strong bisimilarity relied definition depth properties satisfied take account strong bisimilarity particular proved strongly bisimilar processes depth weak bisimilarity property consider following processes notice despite depth depth depth avoid problem adapt ideas behind results previous section consider processes without stuttering transitions transition stuttering transition could establish upd normed processes strong setting norm arguments parallel composition necessarily less norm parallel composition weak setting known normed processes satisfy upd bisimilarity consider following counter example define normed unique parallel decomposition study processes without stuttering transitions section using results developed section theorem section prove finite processes unique parallel decomposition weak bisimilarity processes without stuttering steps write processes implies definition process process without stuttering transitions denote set processes without stuttering transitions lee luttik section discussed consider infinite processes discussion also applies weak bisimilarity definition fact lemma ensure use processes define properties equivalence classes processes weak bisimilarity prove lemma need introduce notation lemma write denote call prefix use range prefixes including lemma prefixes processes proof proof proceeds structural induction cases straightforward definition case induction hypothesis processes bisimulations easy see bisimulation case straightforward induction expansion lemma lemma thanks lemma state case proceeds structural induction three cases analyse denoted iii notice induction processes point proof follows case finally case reduced previous case using expansion lemma lemma lemma every process proof proof result follows complete induction depth lemma prefixes processes induction lemma define get restrict attention processes property executing stuttering transitions preserved parallel composition consider processes compare fact strong setting say possible prove lemma similar lemma processes section conclude collection theorems lemmas theorems equivalent respectively theorems processes weak bisimilarity lemmas needed prove results used next section lemma let proof suppose obtained removing actions therefore contradiction unique parallel decomposition lemma executes least transition proof result straightforward def transition since implies stuttering transition contradicts theorem depth depth proof proceed complete induction depth depth moreover taking account fact creates contradiction stuttering transition therefore depth depth suppose depth let length lemma induction depth depth therefore depth depth prove assume depth reach contradiction follows depth assume depth let length depth lemma depth depth complete induction depth depth reach contradiction depth depth lemma proof let largest sequence hand therefore lemma proof therefore transitions executed stuttering transitions contradicts theorem depth depth depth depth proof prove depth depth proof depth depth analogous note since depth reachable exists remark symbol lemma lemma furthermore convention symmetric version rule lemma imply whenever depth depth given implies lemma addition depth depth finally depth depth depth depth lemma lee luttik unique parallel decomposition development section similar development section reason cases use notation problem developments independent order use theorem need define commutative monoid decomposition order commutative monoid defined notice ensure depth depth extend notion depth following way define depth depth definition sound lemma theorem lemma neutral element binary operation commutative monoid satisfies associativity commutativity identity properties shall define partial order using relation defined follows partial order defined inverse closure write notice case takes account processes weak transitions execute least one transition also notice avoiding communications arguments parallel composition order avoid scope extrusion similarly strong setting need two lemmas lemmas prove partial order lemma addition prove decomposition order need lemma equivalent lemma proofs results follow similarly respective counterpart strong setting complete proofs appendix lemma lemma depth depth lemma partial order lemma depth ready prove decomposition order proof present changes proof lemma except proving archimedean use theorem notice lemma equivalent lemma weak setting lemma decomposition order theorem follows unique decomposition corollary commutative monoid unique decomposition unique parallel decomposition final remarks paper proved finite processes satisfy upd strong bisimilarity weak bisimilarity obtained results using technique presented see theorem different properties satisfied setting strong setting prove properties related depth processes weak setting prove properties related processes execute stuttering transitions results show abstract framework used context dealing complications arise scope extrusion addition framework used deal weak setting one considers processes without stuttering transitions way avoided abstract technique introduced considerably involved technique used paper section showed two examples norm additive therefore proofs flawed pointing problem dreier proposed alternative definition norm solve call variant roughly consider traces scope extrusion processes think solution may work applied suitable variant considered present paper first explain problem context problem present applied first example section process normed goes finite trace process executes scope extrusion consider process would normed according alternative definition suggested result scope extrusion since expansion clear thus find property normed compatible bisimilarity since applied include construct choice needed expansion problem present open question leaves paper related upd strong full bisimilarity strong full bisimilarity stronger notion bisimulation congruence constructs tried apply abstract technique setting far without success tried repeat result section taking account universal quantification definition strong full bisimilarity problem arose wanted prove order decomposition order particularly able prove order strict compatible notice problem present asynchronous fragment strong bisimilarity strong full bisimilarity coincide acknowledgement authors thank daniel hirschkoff discussions comments suggestions various drafts paper anonymous reviewers thorough reviews good suggestions references aceto fokkink luttik ccs hennessy merge finiteequational axiomatization theor comput sci aceto fokkink luttik finite equational base ccs left merge communication merge acm trans comput log lee luttik aceto luttik van tilburg finite equational bases fragments ccs restriction relabelling ifip world computer congress foundations computer science castellani bisimulations concurrency thesis university edinburgh also published christensen decidability decomposition process algebra thesis university edinburgh corradini gorrieri marchignoli towards parallelization concurrent systems informatique applications dreier ene lafourcade lakhnech existence decidability unique theor comput sci decompositions processes applied dreier lafourcade lakhnech defining privacy weighted votes single coercion esorics fokkink luttik equational specification interleaving automata languages programming international colloquium icalp friso groote moller verification parallel systems via decomposition concur third international conference concurrency theory hirschkoff pous distribution law ccs new congruence result calculus logical methods computer science hirshfeld jerrum bisimulation equivalence decidable normed process algebra automata languages programming international colloquium icalp lanese sangiorgi schmitt expressiveness decidability higherorder process calculi inf comput luttik unique parallel decomposition branching weak bisimulation semantics theor comput sci luttik van oostrom decomposition orders another generalisation fundamental theorem arithmetic theor comput sci milner moller unique decomposition processes theor comput sci moller axioms concurrency thesis university edinburgh moller importance left merge operator process algebras automata languages programming international colloquium moller nonexistence finite axiomatisations ccs congruences proceedings lics sangiorgi walker theory mobile processes cambridge university press new york usa
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jan modules infinite regularity commutative graded rings luigi ferraro abstract work prove graded commutative algebra field koszul denoting maximal homogeneous ideal nonzero modules form infinite castelnuovomumford regularity also prove complete intersections koszul nonzero direct summand syzygy infinite regularity finally relate vanishing graded deviations nonzero direct summand syzygy finite regularity introduction let polynomial ring graded setting deg let homogeneous ideal denote maximal homogeneous ideal size minimal free resolution graded rmodule measured graded betti numbers rankk exti invariants arising betti numbers projective dimension regularity avramov eisenbud prove regularity finite regularity every finitely generated module finite avramov proves nonzero modules form infinite projective dimension provided regular article prove nonzero modules form also infinite regularity provided koszul section prove nonzero direct summand syzygy infinite regularity provided complete intersection koszul ask whether holds true ring section provide connection nonzero direct summand syzygy finite regularity vanishing graded deviations tightly embeddable modules section construct class modules infinite regularity rings definition let graded say tightly embeddable exists finitely generated graded case tight embedding follows nakayama lemma finitely generated tight embedding zero luigi ferraro motivated results explore relation tightly embeddable modules regularity establish convention graded space hilbert series graded space rankk bigraded series finitely generated graded let write formal power series every graded graded sth shift graded defined denotes degree remark accordance bigraded version gulliksen levin algebra extr graded hopf algebra universal enveloping algebra graded lie algebra first degree homological second internal degree follows bigraded versions theorems characteristic theorem positive odd characteristic theorem characteristic notation denote algebra extr every denote left extr following theorem graded version lemma similar proof theorem tight embedding inequality pkr sdi proof set following notation consider following commutative diagram exists modules infinite regularity commutative graded rings induces commutative diagram homomorphisms bigraded left annihilates natural isomorphisms let denote universal enveloping algebra consider following commutative diagram vertical maps natural injective maps fix map also hence theorem free basis hence free basis means generated elements change basis coordinate vectors respect basis linearly independent zero elements part deduce linearly independent result means contains copy based commutativity diagram gives luigi ferraro according equality formal power series every deduce consider following chain equalities hilbert series first equality follows exact sequence cohomology induced first row second equality follows already mentioned fact every third equality follows second row fourth equality follows last equality follows holistically obtain observe pkr hence obtain replacing obtain desired inequality let finitely generated graded regularity reg sup definition ring koszul algebra extr generated according reg reg result happen koszul modules infinite regularity commutative graded rings corollary tightly embeddable module reg koszul proof set reg reg sdi theorem pkr deduce equivalent reg direct summands syzygies residue field section prove nonzero direct summands syzygies infinite regularity complete intersection generated quadrics special homological properties class modules already noticed theorem complete intersection rmodule homomorphism finitely generated graded map extnr extnr zero inequality pkr sdi proof set following notation let canonical projection morphism obtained extending morphism free resolutions let denote composed map iterated connecting homomorphism hence isomorphism construction result theorem graded left complete intersection see example theorem theorem references given polynomial ring hence submodule free module polynomial ring result nonzero element image generates copy whose internal degree might shifted follows conclude theorem corollary let complete intersection reg nonzero direct summand infinite regularity luigi ferraro raise following question question homomorphism finitely generated graded map extnr extnr zero finite regularity koszul vanishing graded deviations section relate question vanishing graded deviations let rhxi acyclic closure see definition graded rings require differential acyclic closure homogeneous map differential homogeneous must give elements internal grading making bigraded set definition graded deviation card set variables acyclic closure homological degree internal degree theorem homomorphism finitely generated graded map extnr extnr zero reg proof set following notation let canonical projection morphism obtained extending morphism free resolutions let universal enveloping algebra let denote composed map consider following commutative diagram theorem given odd generated elements modules infinite regularity commutative graded rings set change basis coordinate vectors respect basis linearly independent zero elements form basis deduce also linearly independent therefore means contains copy construction recall theorem dimk even nonzero element powers belong however copy contained bideg bideg goes goes implies reg contradiction infinitely many odd infinitely many nonzero belonging products belong however copy contained bidegree product bideg every following inequality holds goes obtain reg contradiction assume finitely many odd let rhxi acyclic closure choose element odd degree maximal homological degree denote homological degree homogeneous element acyclic closure deg internal degree claim subalgebra acyclic closure indeed odd elements lemma subalgebra acyclic closure maximality subalgebra acyclic closure hence algebra acyclic closure written ahyi implying lemma contradiction proposition let even number luigi ferraro proof let rhxi acyclic closure let element homological degree need prove deg assume appears boundary homogeneous element rwy following string equalities observed deg deg deg deg deg deg deg deg deg deduce deg deg deg deg possible hence deg appear boundary element write acyclic closure ahyi lemma hence possible set known see theorem motivated previous proposition raise following question question true theorem theorem regular complete intersection theorem ring form regular standard graded koszul ring motivated results raise following question question ring form complete intersection standard graded koszul ring acknowledgements author would like thank advisors avramov iyengar helpful conversations references hopf algebras divided powers algebra avramov infinite free resolutions six lectures commutative algebra bellaterra progr math basel avramov modules extremal resolutions math res lett avramov eisenbud regularity modules koszul algebra algebra avramov peeva finite regularity koszul algebras amer math gulliksen levin homology local rings queens papers pure appl math queens kingston modules infinite regularity commutative graded rings martsinkovsky remarkable property syzygy modules residue field nonregular local ring pure appl algebra milnor moore structure hopf algebras ann math hopf algebras derivations algebra takahashi syzygy modules semidualizing summands algebra
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feb counting uniform sampling markov equivalent dags amiremad saber negar department ece coordinated science laboratory university illinois urbana usa electrical engineering department sharif university technology tehran iran kiyavash saleh propose exact solution problem finding size markov equivalence class mec bounded degree graphs proposed solution capable computing size mec polynomial time proposed approach based recursive method counting number elements mec specific vertex set source variable use idea design sampler capable sampling mec uniformly polynomial time introduction directed acyclic graphs dags commonly used graphical model represent causal relationships among set variables dag representation directed edge indicates direct causal relationship corresponding variables markov property faithfulness assumptions conditional variables dag bijective correspondence conditional independencies variables underlying joint probability distribution spirtes hence dag representation demonstrates conditional independencies among variables general approach learning causal structure use statistical data variables find dag consistent conditional independencies given data however dag representation set conditional independencies always unique restricts learning causal structure markov equivalence classes mecs elements class represent set conditional independencies preference elements mec therefore given set data generally given joint probability distribution variables size mec number elements clear metric measuring success observational structure learning approach indicates necessity using interventional experiments indicates importance awareness size mec given structure interest moreover mentioned earlier one compare different dags find consistent one given joint distribution variables space dags space mecs chosen search space chickering showed size mecs large searching space mecs efficient chickering indicates importance learning size mecs best knowledge main existing solution problem finding size mecs use markov chain methods bernstein tetali according method markov chain constructed elements mec whose properties ensure stationary distribution uniform elements main problems markov chain method time required convergence computational issues cases prevents use practice paper propose exact solution problem finding size mec bounded degree graphs proposed solution capable computing size mec polynomial time proposed approach based recursive method counting number elements mec specific vertex set source variable mentioned earlier observational setting preference elements mec therefore reasonable approach testing performance algorithm mec test uniformly drawn samples mec indicates importance ability sampling mec uniformly use introduced idea design sampler capable sampling mec uniformly polynomial time approach applied finding best set variables intervene restricted certain budget number experiments ghassami rest paper organized follows section present problem description method counting elements mec presented section counting idea used section design uniform sampler markov equivalent dags problem description mec usually represented mixed graph graph containing directed undirected edges denote set vertices set directed edges set undirected edges respectively directed edges direction elements class directions essential reason mixed graphs referred essential graph literature andersson rest edges undirected exist two elements class agree direction edge essential graphs also called complete partial directed acyclic graphs pdags chickering maximally oriented graphs meek define skeleton dag undirected graph set vertices original dag undirected version edges dag also define structure comprising two converging directed edges whose tails connected edge following result due verma pearl verma pearl characterizes markov equivalence proposition verma pearl two dags markov equivalent skeleton proposition indicates edges part directed essential graph none members mec subgraph corresponding undirected part essential graph allowed contain let essential graph consider undirected graph resulted removing directed edges graph consists disconnected chain components note components essential graph entire essential graph undirected forms single chain component andersson showed chain components chordal graphs andersson let size denote size mec corresponding gillispie perlman showed size calculated follows gillispie perlman size size result suggests one study chain component separately therefore study problem finding size mec essential graph connected undirected chordal graph hence none members mec allowed counting members mec goal section count members mec essential graph connected undirected chordal graph define source variable digraph one incoming degree zero also let distance vertices following result bernstein tetali used proposed method proposition bernstein tetali acyclic orientation connected graph unique source variable unique source determines orientation edges proposition determines direction edges resolved source vertex known edges one endpoint closer source vertex note removing resolved edges obtained digraph remain acyclic lemma original digraph acyclic source vertex graph resulting removing edges also acyclic proof acyclicity obvious removing edges show consider note created means shielding vertex removed since unresolved contradiction due lemma orient edges obtain member corresponding mec follows first one vertex chosen source vertex determine orientation edges according proposition removing resolved edges resulted isolated vertices possibly disconnected components components contain cycles component one vertex chosen source vertex component procedure repeated unresolved edge left idea aforementioned procedure used recursive manner count members mec main tool approach presented theorem theorem let number elements mec connected undirected essential graph source vertex let undirected graph remained removing edges one endpoint closed well removing resulting isolated vertices comp denotes set components convention empty product nullary product assumed multiplicative identity proof component choosing different variables component source partitions set elements mec two different choices source lead dag due proposition elements remained uncounted variable chosen source therefore inner summation justify outer product need show orientation one component cause restrictions possible orientations component due fact component certain depth level source create pointing lower levels also reason fact edges source towards component edges two component create cycle therefore using theorem order count members mec suffices calculate sum size approach presented algorithm code consists two functions lowed function lowed takes undirected connected graph one vertices input returns set edges whose orientation resolved input vertex set source vertex function applies theorem recursive manner calculate number elements mec connected undirected essential graph input vertex set source vertex theorem let denote order maximum degree graph respectively computational complexity algorithm proof consider operation algorithm rounds first round lowed called pairs specific obtain undirected components second round component algorithm mec size calculator input undirected graph size output size function lowed initiate set source variable end end end return end function function undirected version elements lowed remove isolated vertices return end function call lowed pairs specific obtain undirected components continue procedure suffices show procedure could repeated rounds therefore since round function lowed called times get desired result suppose round obtain undirected component execute lowed pair let resulting undirected components first show vertex obtained directed edge vertex applying lowed contradiction suppose directed edge consider shortest path path must pass one neighbors since oriented calling lowed procedure contradiction fact thus directed edge vertex consider set edges intersect figure graphs related example based reasoning least one edges oriented round vertex still one undirected components round since degree edges oriented rounds example consider undirected graph presented figure essential graph given input algorithm setting vertex source vertex direction edges obtained using function lowed shown figure therefore remaining undirected graph shown figure setting vertex source vertex undirected component remained hence therefore due symmetry structure well setting vertex source vertex direction edges obtained using function lowed shown figure therefore remaining undirected graph shown figure setting vertex source vertex undirected component remained hence therefore due symmetry structure well finally using equation obtain number elements mec corresponding essential graph equal algorithm uniform sampler input undirected graph output directed graph dge function dge set source variable probability lowed undirected version elements lowed remove isolated vertices dge return end function uniform sampling mec mentioned earlier section applications performance action needed tested mec large size cases one optimize behavior uniform samples drawn mec section introduce two samplers providing random dags mec first sampler presented subsection based counting idea section case bounded degree graphs sampler capable producing uniform samples polynomial time degree bounded worst case sampler high computational complexity therefore subsection provide another sampler originally introduced ghassami case unbounded degree graphs although sampler unbiased extensive experimental results confirms sampling distribution second sampler close uniform uniform sampler uniform sampler presented algorithm functions lowed defined algorithm sampler first weights calculated vertices vertex chosen source variable probability source variable determine orientation edges according proposition removing directed edges resulted isolated vertices possibly disconnected components choosing source variable resulting component repeated edges directed since choose vertex source variable according portion members mec vertex source following result theorem sampler proposed algorithm uniform proof let member mec clarity proof assume obtained form three rounds choosing source vertices otherwise idea easily extendable case arbitrary many rounds therefore assume obtained chosen source vertex round sources components round sources components round since process terminated three rounds therefore theorem second fraction fractions product equal therefore size last equality follows equation regarding complexity sampler algorithm following result corollary theorem corollary let denote order maximum degree graph respectively computational complexity algorithm algorithm heuristic sampler input undirected graph uniformly shuffle order elements induced subgraph subset size variables directed directed cycle orient undirected edges among independently according bern becomes directed structure directed cycle end end output resulted directed graph heuristic sampler case unbounded degree computational complexity algorithm high worst case therefore also provide heuristic sampler case first introduced ghassami although unbiased experimental results indicate sampler performance close uniform sampler heuristic sampler presented algorithm algorithm consider subsets size uniformly random order achieved uniformly shuffling labels elements subset orient undirected edges among independently according bernoulli distribution resulting orientation induced subgraph became directed cycle redo orienting keep checking subsets size induced subgraph directed none directed cycle theorem generated directed graph algorithm belongs markov equivalence class corresponding essential graph require following lemma proof lemma chordal graph directed cycle directed cycle size proof directed cycle size lemma trivial suppose cycle size relabel vertices since graph chordal chord hence triangle vertices direction directed cycle size otherwise directed cycle vertices relabeling vertices repeating reasoning concludes lemma proof theorem undirected graph chordal hauser therefore lemma insure generated directed graph dag suffices make sure directed cycles length one checks proposed procedure checking generated dag markov equivalence class corresponding proposition suffices check set check proposed procedure references notes andersson andersson madigan perlman characterization markov equivalence classes acyclic digraphs annals statistics bernstein tetali bernstein tetali sampling graphical markov models arxiv preprint chickering chickering optimal structure identification greedy search journal machine learning research nov ghassami ghassami salehkaleybar kiyavash bareinboim budgeted experiment design causal structure learning arxiv preprint gillispie perlman gillispie perlman size distribution markov equivalence classes acyclic digraph models artificial intelligence hauser hauser characterization greedy learning interventional markov equivalence classes directed acyclic graphs journal machine learning research aug jia reversible mcmc markov equivalence classes sparse directed acyclic graphs annals statistics meek meek causal inference causal explanation background knowledge proceedings eleventh conference uncertainty artificial intelligence pages morgan kaufmann publishers spirtes spirtes glymour scheines causation prediction search mit press verma pearl verma pearl equivalence synthesis causal models proceedings sixth conference uncertainty artificial intelligence pages
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apr fast hybrid primal heuristic multiband robust capacitated network design multiple time fabio andreagiovannia jonatan krolikowskia jonad pulaja department optimization berlin zib takustr berlin germany dfg research center matheon technical university berlin des juni berlin germany einstein center mathematics berlin ecmath des juni berlin germany abstract investigate robust multiperiod network design problem generalization capacitated network design problem cndp besides establishing flow routing network capacity installation canonical cndp also considers planning horizon made multiple time periods protection fluctuations traffic volumes remedy traffic volume uncertainty propose robust optimization model based multiband robustness andreagiovanni refinement classical bertsimas sim uses system multiple deviation bands since resulting optimization problem may prove challenging even instances moderate size solved authors final version paper published applied soft computing doi final publication available elsevier sciencedirect via http please note paper extended improved journal version paper hybrid primal heuristic robust multiperiod network design presented published evoapplications applications evolutionary computation granada spain april lncs springer doi paper awarded evocomnet best paper award evostar corresponding author email addresses fabio andreagiovanni krolikowski jonatan krolikowski pulaj jonad pulaj url http fabio andreagiovanni optimization solver propose hybrid primal heuristic combines randomized fixing strategy inspired ant colony optimization exact large neighbourhood search computational experiments set realistic instances sndlib show original heuristic run fast produce solutions extremely high quality associated low optimality gaps keywords capacitated network design multiperiod design multiband robust optimization traffic uncertainty metaheuristic ant colony optimization exact large neighborhood search introduction last two decades telecommunications increasingly pervaded everyday life volume traffic sent exchanged networks astonishingly increased major companies like nokia siemens networks expect increase amount traffic strongly continue reaching volume exabyte per year fixed networks theimer dramatic growth telecommunications experienced greatly compounded challenge network professionals facing design problems increasing complexity difficulty order cope traffic growth professionals plan much advance network expanded topology capacity accommodate new traffic especially important case fixed networks require costly digging operations installation cables areas possibly high population density make design task even complicated future behaviour traffic network exactly known network designed thus decision problem also affected tricky data uncertainty recent times data uncertainty generally neglected real studies however indicated recent industrial cooperations industry academia bauschert belotti bley koster professionals becoming aware importance adopting mathematical optimization take better decisions also understanding necessity considering data uncertainty order avoid unpleasant surprises like infeasibility implemented solutions due data deviations task designing telecommunication network essentially consists establishing topology network technological features transmission capacity rate elements namely nodes links one studied problem network design capacitated network design problem cndp cndp consists minimizing total installation cost capacity modules network route traffic flows generated users cndp central problem network optimization appears many applications exhaustive introduction refer reader ahuja bertsekas bienstock paper focus development new robust optimization model tackle traffic uncertainty multiperiod capacitated network design problem problem constitutes natural extension classical cndp instead single design period consider design time horizon made multiple periods moreover traffic uncertainty taken account protect design solutions deviations traffic input data may compromise feasibility optimality solutions immediately stress though problem optimally designing networks multiple time periods new traced back least seminal work christofides brooker best knowledge received little attention works investigated essentially lardeux gavros raghavan checking literature several works dealing multiperiod design networks found make couple examples gendrau gupta grossmann however works consider problems sensibly different general setting consider thus avoid detailed discussion gendrau study capacity expansion problem access networks tree topology whereas gupta grossmann consider design utility networks modeled mathematical programs main references work namely lardeux gavros raghavan point difficulty solving multiperiod cndp problems already two periods even easier contexts lardeux consider cndp traffic flows may split whereas gavros raghavan consider pure routing problem satellite communications direct recent computational experience confirmed challenging nature even instances moderate size low number time periods solved optimization solver uncertain versions traffic uncertainty considered also neglected even though especially last years increasing interest network design traffic uncertainty single design period case bauschert belotti koster work main original contributions first robust optimization model tackling traffic uncertainty multiperiod cndp specifically adopt multiband robustness new model robust optimization recently introduced andreagiovanni hybrid primal heuristic based combination randomized rounding heuristic resembling ant colony optimization dorigo exact large neighborhood search called rins danna stress aim use standard implementation ant colony algorithm wanted instead strengthen performance ant algorithm using highly valuable information linear relaxations considered optimization problems using information allowed define strong ant construction phase produces high quality solutions already execution local search analytically proving solve linear relaxation multiperiod cndp closed form thus obtaining substantial reduction solution times first algorithm presented andreagiovanni computational experiments set realistic instances derived survivable network design library sndlib showing hybrid algorithm able produce solutions extremely high quality associated small optimality gap remainder paper organized follows section review canonical model cndp section introduce multiperiod cndp study linear relaxation section introduce new formulation robust multiperiod cndp sections present hybrid heuristic computational results capacitated network design problem cndp central highly studied problem network optimization appears wide variety applications see ahuja bertsekas exhaustive introduction essentially described follows given network set demands whose flows must routed vertices network want install capacities network edges route flows network capacity constraint edge respected total cost installing capacity minimized formally characterize cndp following definition definition capacitated network design problem cndp given network represented graph set vertices set edges set commodities associated traffic flow route origin destination set admissible paths routing flow commodity cost installing one module capacity edge cndp consists establishing number capacity modules installed edge resulting capacity installation minimum cost supports feasible routing commodities feasible routing assigns commodity exactly one feasible path referring notation introduced introducing following two families decision variables binary path assignment variables xcp entire traffic commodity routed path xcp otherwise integer capacity variables representing number capacity modules installed edge model cndp following integer linear program min xcp xcp xcp objective function minimizes total cost capacity installation capacity constraints impose summation flows routed edge must exceed capacity installed equal number installed modules represented multiplied capacity granted single module constraints impose flow commodity must routed single path remark unsplittable version cndp namely traffic flow commodity split multiple paths going must routed exactly one path moreover set feasible paths commodity preset constitutes input problem line works based industrial cooperations bley experience bauschert network operator typically considers paths meet specific business considerations uses routing schemes based open shortest path first protocol multiperiod capacitated network design problem define multiperiod generalization cndp introducing time horizon made set elementary periods modeling point view generalization requires add new index decision variables represent routing capacity installation decisions taken period simple modeling operation however greatly increases size complexity problem pointed computational section gavros raghavan lardeux introduction new index leads new integer linear program min yet xtcp dtc xtcp xtcp yet besides including new index program presents modified capacity constraint time period must consider presence capacity modules installed edge period period concerning costs traffic demands follows realistically assume cost per unit capacity time demand associated commodity time dtc computing linear relaxation introduced problem proceed formally characterize value optimal solution linear relaxation namely problem obtain relax integrality requirements variables xtcp yet refer relaxation characterizing optimal value structure optimal solution crucial objective preliminary computational experience presented proceeding version paper andreagiovanni solving using optimization solver ibm ilog cplex proved indeed slow using solver like cplex solve linear relaxations thus constituted bottleneck limited possibility running heuristic large number times within time limit using instead expression coming new theoretical results presented section able efficiently solve without using cplex run algorithm incredibly higher number times time limit gain determinant speed overall execution algorithm case including single time period well known optimal solution linear relaxation obtained considering shortest path commodity installing edge smallest number capacity modules needed support traffic request see ahuja following propositions investigate optimal value optimal solution look like proving characterized efficiently note order keep exposition light decided move complex proofs statements appendix first step characterize relation flows consecutive time periods proposition let dtcp exists optimal solution dtc proof see appendix using previous proposition able derive following theorem characterizes optimal solution linear relaxation particular show get optimal solution need identify commodity time period shortest path edge length among feasible paths course operation done fast efficiently remark stress follows focus determination optimal flows since optimal flow capacity installation immediately derived network theorem consider problem namely linear relaxation let shortest path edge length commodity optimal solution defined routing time period entire flow commodity shortest path installing edge exact capacity needed route traffic flow dtc proof see appendix previous theorem efficiently characterizes optimal solution however hybrid algorithm requires feasible solution linear relaxation routing capacity installation established number consecutive time periods see section details following corollary shows characterize feasible solution modified linear relaxation identifies conditions solution becomes optimal corollary consider problem suppose time periods commodities feasible path assignments capacity installations established thus fixing values corresponding variables moreover suppose period feasible path assignments capacity installations established subset commodities denote version obtained variable fixing consider corresponding linear relaxation let shortest path edge cost commodity feasible solution defined routing time period entire flow shortest path installing edge minimum number capacity modules needed route traffic flow routing time period entire flow shortest path installing edge minimum number capacity modules needed route traffic flow dtc furthermore solution determined optimal commodities path chosen every proof feasibility solution built specified clear optimality condition instead straightforward consequence theorem omit proof new theoretical results introduced provide alternative way compute optimal feasible solution prove dramatically faster direct use cplex allows greatly increase number executions ant construction phase time limit robust optimization multiperiod network design introduced multiperiod generalization proceed consider version takes account traffic uncertainty end section first state mean traffic uncertainty present fundaments robust optimization methodology adopt tackle data uncertainty finally present robust optimization model version traffic uncertainty robust optimization uncertainty traffic naturally present telecommunications network design since future behaviour customers known advance number users traffic generated estimated estimates deeply differ actual traffic conditions occur future see bauschert follows thus assume demands uncertain commodities value known exactly optimization problem solved order clarify concept traffic uncertainty anticipate model data uncertainty refined interval deviation model interval model assume know nominal value traffic maximum negative positive deviations unknown actual value thus assumed belong interval direct experience network design observed professionals often identify value forecast traffic volume expected value derived historical data whereas deviations identified maximum deviations forecast considered relevant network designer using historical data reference example traffic uncertainty consider two commodities associated nominal traffic demands suppose values may deviate maximum negative positive deviations respectively actual values traffic therefore sensitivity analysis dealing data uncertainty optimization problems may result really tricky small variations value input data may completely compromise optimality feasibility produced solutions solutions supposed optimal may reveal heavily suboptimal whereas solutions supposed feasible may reveal infeasible thus meaningless implemented detailed discussion issues associated data uncertainty refer reader bertsimas following example immediately help visualize possibly catastrophic effects neglecting data uncertainty example infeasibility caused deviations consider commodities example suppose link installed exactly capacity handle sum nominal values installed capacity capacity installation neglects fact demands may deviate sufficient one commodity experiences positive deviation violate capacity constraint link thus making design solution infeasible practice previous example makes clear afford neglect traffic uncertainty therefore risk design solution turn infeasible bad quality implemented consequence decided tackle data uncertainty adopting robust optimization approach methodology dealing data uncertainty received lot attention recent times preferred traditional methodologies like stochastic programming especially thanks accessibility computational tractability refer reader bertsimas exhaustive introduction theory applications discussion determinant advantages stochastic programming founded two main facts decision maker must define uncertainty set reflects risk aversion identifies deviations coefficients protection must guaranteed protection deviations included uncertainty set guaranteed hard constraints cut feasible solutions may become infeasible deviations uncertainty set formal way suppose given generic integer linear program max subject whose coefficient matrix uncertain namely know exact value entries however identify family coefficient matrices constitute possible valorizations uncertain matrix family uncertainty set robust problem robust optimal solution solution protected data deviations found considering following robust counterpart original problem max subject feasible set robust counterpart includes solutions feasible coefficient matrices uncertainty set consequently subset feasible set original problem stress definition robust counterpart extended linear program involving continuous integer variables time additionally remark decision maker include coefficient matrices reflect specific risk aversion ensuring protection data deviations according paradigm comes price price robustness por bertsimas sim por deterioration optimal value robust counterpart optimal value original problem caused hard constraints imposing robustness restrict feasible set general smaller set robust solutions por depends upon features uncertainty set uncertainty sets reflecting higher levels risk aversion decision maker include unlikely extreme deviations leading higher protection yet associated higher por uncertainty sets reflecting low risk aversion instead tend neglect unlikely deviations thus guaranteeing lower protection yet associated lower por example protection deviations following example simple way grant protection would install sufficient capacity deal peak deviations commodity install capacity note practice unlikely coefficients experience worst deviation one aim smart models define appropriate uncertainty sets result conservative guaranteeing satisfying level protection example could assume one two demands deviate nominal value next paragraph provide description model uncertainty adopt concise introduction multiband robust optimization work tackle uncertainty multiband robust optimization new robust optimization model based cardinalityconstrained uncertainty set proposed andreagiovanni extended applied series successive works andreagiovanni bauschert represents refinement generalization bertsimas sim developed satisfy practical needs industrial partners applications see andreagiovanni bauschert recall main results referring following generic uncertain linear program milp max ilp aij assume uncertainty affects coefficients aij uncertainty affecting cost coefficients easily reformulated coefficient matrix uncertainty moving hypothesis actual value aij coefficients unknown multiband uncertainty model basis assumes coefficient aij decision maker knows nominal value well maximum negative positive deviations dij aij dij overall single deviation band dij coefficient aij partitioned bands defined basis deviation values dij dij dij dij deviation values deviation bands defined namely set positive deviation bands set negative deviation bands band corresponds range dij dij band corresponds single value constraint band lower bound lik upper bound uik number deviations may fall defined lik uik number coefficients take nominal value limited lik always exists feasible realization coefficient matrix call typology uncertainty set multiband uncertainty set thus generalizes uncertainty definition model bertsimas sim single deviation band partitioned multiple bands band associated upper bound uik also lower bound lik number coefficients deviating band lower bound improves modeling power decision maker importantly allows take account presence negative value deviations neglected course taking account negative deviations allows improve modeling deviations commonly found problems even critically reduces value overall worst deviation thus price robustness multiband model results particularly suitable model histograms commonly adopted practitioners visualize analyze data deviations see andreagiovanni bauschert since robust optimization paradigm entails must protected possible deviation considered uncertainty set robust counterpart milp multiband uncertainty max devi additional term devi introduced every feasibility constraint represent maximum total deviation could incurred constraint multiband uncertainty set solution problem actually since term devi hides following maximization problem devi max dkij yij lik yij uik yij yij problem equal coefficient binary variable yij constraint falls deviation band equal otherwise coefficient constraint must fall one deviation band thus requiring introduction family constraints note like coefficient falling zero deviation band constraints impose bounds number coefficients may deviate band finally objective function aims maximizing deviation allowed multiband uncertainty set given solution constraint robust counterpart problem since includes binary program anyway real issue since proved robust counterpart equivalent compact linear linear program stated following theorem theorem andreagiovanni robust counterpart problem milp multiband uncertainty set equivalent following compact linear program max wik wik zij dkij wik zij zij problem includes additional continuous variables additional constraints clear trivial robust counterpart moreover constraints include additional terms involve values express number deviations occur deviation band values constitute profile multiband uncertainty set derived basis bounds see andreagiovanni details resulting formulation thus nice properties compact linear proof theorem based pointing integrality polyhedron associated exploiting strong duality refer reader andreagiovanni formal complete statement proof presented theorem theorem central result theory multiband robust optimization use derive robust model multiperiod network design proceed use multiband robust optimization related theorem tackle traffic uncertainty denote uncertainty set associated demands commodities write general form robust counterpart follows min yet xtcp xtcp devet xtcp yet robust counterpart differs capacity constraints constraints indeed consider nominal traffic demands values include terms devet represent total maximum positive deviation demands may experience edge period routing vector uncertainty set structure uncertainty set according principles multiband robust optimization introduced previous subsection specifically build multiband uncertainty set follows commodity time period know nominal value traffic coefficient maximum negative positive deviations actual value dtc dtc overall deviation range coefficient dtc partitioned bands defined basis deviation values deviation values deviation bands defined namely set positive deviation bands set negative deviation bands band corresponds range dtk band corresponds single value dtk capacity constraint defined edge period band introduce two values letk utk represent lower upper bound letk utk number traffic coefficients whose value deviates band nte number uncertain coefficients constraint bounds used derive profile uncertainty set namely values indicating exact number coefficients deviating band see andreagiovanni details definition profile number coefficients take nominal value limited always exists feasible realization coefficient matrix using previous characterization multiband uncertainty set theorem reformulate robust counterpart following linear compact robust counterpart min yet xtcp wetk zecp wetk zecp xtcp wetk zecp xtcp xtcp yet formulation includes additional constraints variables linearly reformulate original problem including term devet capacity constraint problem want solve computational section get robust solutions hybrid primal heuristic principle get robust optimal solution using commercial programming software cplex however showed computational section solving constitutes difficult task even considering small number time periods using solver like cplex several hours computation solutions still typically low quality far away optimum remedy attracted effectiveness hybrid heuristics hard network design problems valid examples effectiveness provided crainic gendreau proposing cooperative parallel tabu search algorithm singleperiod cndp tested transshipment networks dely proposing linear decomposition method variant cndp related fair routing wireless mesh networks kleeman proposing multiobjective evolutionary algorithm solve cndp arising design large communications networks concerning hybrid heuristics refer reader blum recent survey case developed fast hybrid primal heuristic based combination randomized strategy resembling ant colony optimization aco exact large neighbourhood search widely known aco metaheuristic inspired foraging behaviour ants seminal work dorigo presenting aco algorithm combinatorial problems later extended integer continuous problems dorigo followed hundreds papers proposing refinements basic algorithms gambardella maniezzo investigating applications relevant optimization problems see blum overview aco algorithm presents general structure specified algorithm loop consisting two phases executed arrest condition satisfied first phase ant builds solution guidance probabilistic functions variable fixing resemble pheromone trails pheromone trails updated basis effective adopted variable fixing resulted arrest condition reached daemon action phase takes place solution improvement strategy applied bring feasible solution built ants local optimum algorithm general structure aco algorithm arrest condition reached solution construction pheromone trail update end daemon actions proceed detail phase previous sketch hybrid algorithm algorithm defined hybrid since passed aco construction phase daemon action phase operates exact large neighborhood search formulated integer linear program solved exactly relying power modern commercial solvers solution construction first step inner cycle number ants defined ant iteratively builds feasible solution problem generic iteration construction ant state corresponding partial solution make step towards completing solution making move move corresponds fixing value variable move executed chosen according probability function function specifies probability implementing move thus fixing variable derived combining priori measure efficacy move posteriori measure efficacy move detail probability choosing move fixes variable fixed variable specified following canonical formula pij represents priori measure efficacy commonly called pheromone trail value represents posteriori measure efficacy commonly called attractiveness note formula influenced two parameters appear exponents measures chosen decision maker basis specific problem considered detailed description elements actions phase refer reader paper maniezzo presents ants improved ant algorithm used solving quadratic assignment problem taken reference work considered ants particularly attractive proposes series refinements classical aco allow better exploit polyhedral information problem specifically maniezzo sketches ideas alternative formulations original problem could exploited define pheromone trail attractiveness values additional desirable feature ants makes use reduced number parameters adopts efficient mathematical operations canonical ant algorithms products instead exponentiations exhaustive description ants refer reader paper maniezzo describing ants implementation structured make preliminary considerations formulation based four families variables path assignment variables xtcp coming capacity variables yet auxiliary variables wetk zecp robust dualization though deal four families notice routing decisions taken entire time horizon immediately derive capacity installation minimum cost indeed values path assignment variables fixed routing completely established worst traffic deviation term devet efficiently derived without need using auxiliary variables wetk indeed traced back solving flow problem zecp explained andreagiovanni easily derive total traffic det sent edge period worst case derive minimum cost installation sequential evaluation period period keeping mind must det capacity modules accommodate traffic construction phase consequently limit attention binary assignment variables introduce concept routing state definition routing state let let subset triples representing assignment path commodity period routing state assignment paths subset commodities subset time periods excludes multiple paths assigned single commodity formally say routing state complete specifies path used commodity time period thus otherwise called partial ants algorithm propose decided assign paths considering time periods commodities order specifically establish routing time period separately starting continuing time period commodities sorted descending order nominal traffic demand formally operated cycle specified algorithm builds complete routing state iteration nested cycles algorithm assignment path commodity equivalent ant moves partial routing state rsi partial routing state rsj rsj rsi note definition routing state sequence moves actually sequence fixings decision variables maniezzo algorithm construction complete routing state sort descending order sorted assign single path end end probability ant moves routing state complete routing state chosen among set feasible routing states defined improved formula proposed maniezzo pkij parameter assessing relative importance trail attractiveness formula presents two peculiar advantages canonical formula adopts single parameter place two parameters uses coefficient product instead index exponentiation discussed maniezzo trail values attractiveness values provided suitable lower bounds considered optimization problem particular case derived values variables solution associated linear relaxation robust counterpart equal value good feasible solution linear relaxation subset decision variables fixed values effect fixing decision taken previous steps algorithm daemon actions relaxation induced neighborhood search end phase attempt improving quality feasible solution found executing exact local search large neighborhood particular adopt modified relaxation induced neighborhood search rins see danna exhaustive description method integer linear program look solution integral time guarantees best objective value rins observed optimal solution linear relaxation problem provides objective value better feasible solution however time optimal solution fractional therefore guarantee integrality contrary feasible solution guarantees integrality provides worse objective value fact variable fixed value optimal solution linear relaxation feasible solution good indication fixing variable good thus maintained given feasible solution rins profits observations defining search neighborhoods variables value feasible solution optimal solution linear relaxation fixed free vary value neighborhood explored exhaustively formulating search integer linear program solved exactly possibly including arrest condition solution time limit specific case let feasible solution found ant xlr optimal continuous solution linear relaxation modified rins entails solve exactly optimization solver like cplex subproblem fix variables whose value xlr differs xlr xlr impose solution time limit optimization solver time limit imposed since subproblem may difficult solve exploration large neighbourhood may need truncated note point generalize fixing rule rins thus allow fixed variables may differ value contrast canonical rins must exactly value relaxation feasible solution pheromone trail update end phase update pheromone trails move according improved formula proposed maniezzo curr values set using linear relaxation set equal values corresponding optimal decision variables equal optimal value relaxation additionally zcurr value solution built ant moving average values last feasible solutions built formula desirable property replacing pheromone evaporation factor parameter whose setting may result tricky moving average whose setting proved much less critical algorithm shows structure original hybrid algorithm algorithm based execution two nested loops outer loop repeated time limit reached execution inner loop defines ants build solutions pheromone trail updates done end execution inner loop ant construction phase applied try get improvement exact large neighborhood search algorithm hybrid algorithm compute linear relaxation initialize values time limit reached build complete routing state derive complete feasible solution end update according end apply best feasible solution experimental results order assess performance hybrid algorithm executed computational tests set instances based realistic network topologies sndlib defined collaboration industrial partners former ongoing industrial projects see bley bauschert instances consider network topologies whose main features presented table instance table report identification code features vertices edges commodities network topology defined three instances considering distinct number time periods namely performed experiments machine ghz processor table features instances name norway geant france pdh polska ram using commercial solver ibm ilog cplex version instances lead large hard solve observed even solver like cplex big difficulties identifying good feasible solutions majority cases final optimality gap contrast clear table cases hybrid primal heuristic able find high quality solutions associated low optimality gaps optimality gap indicates far best feasible solution found value best lower bound available optimal value formally gap note using cplex essential providing lower bound problem instances thus quality guarantee given solutions case solutions produced cplex computed optimality gap referring lower bound produced cplex basis preliminary tests found effective setting parameters heuristic thus balanced attractiveness trail level ants width moving average equal number ants tolerance fixing minutes time limit imposed execution commodity admits feasible paths deviations bands positive negative null deviation band contrast first computational experience presented andreagiovanni linear relaxations solved exactly cplex requiring amount time compute linear relaxations nominal problems results presented section solve linear relaxation primal simplex method implemented cplex allowed greatly reduce time execution ant construction phase hugely increase number defined ants complete set results presented table show performance hybrid solution approach denoted three measures aco gapar respectively represent value best solution found pure aco value best solution found aco followed rins corresponding final optimality gap moreover show performance cplex denoted measures gapip representing value best solution found corresponding final optimality gap value best solutions found instance highlighted bold type overall time limit execution heuristic hour time limit imposed cplex used solve robust counterpart stress increasing time limit bring remarkable benefit cplex even letting cplex run many hours effects getting negligible improvements best lower bound observed extremely slow improvement rate bound running memory huge size search trees generated case two instances cplex even able find feasible solution within time limit cases denoted best solutions found hybrid algorithm cases value least one order magnitude better found cplex better average excluding course cases cplex find feasible solutions results high quality given low optimality gap suppose solutions actually optimal much higher performance particularly evident instances observe contrast conference paper andreagiovanni rins able improve value best solution found ant construction phase new computations role rins reduced thanks dramatic solving linear relaxations able really implement swarm exploration feasible set getting solutions believe optimal close optimum finding improvement rins impossible really unlikely also think fact powerful local search like rins minor null role finding higher quality solutions indication defined strong effective ant construction phase contrast common experience aco indeed really require final search phase get solution sufficiently high quality focus specific network topology may noticed optimality gaps produced algorithm tend become smaller number time periods grows may result surprising counterintuitive first sight one would expect exactly opposite behaviour yet considering assumption demands commodity reasonable think primal hybrid heuristic discovers converging paths number time periods large thus performing better observe likely behaviour would occur case demand assumption dropped way assess validity effectiveness new heuristic also used benchmark fast easy method define possibly good feasible solution suggested theoretical results presented section time period simply route entire flow commodity shortest path however clear table report value solution found approach corresponding optimality gap gapsp vast majority instances particular instances performance algorithm far better simple approach performed much worse heuristic believe provides evidence solidity algorithm conclusion future work paper introduced first robust optimization model handle uncertainty affects traffic demands multiperiod capacitated network design problem data uncertainty may compromise quality feasibility produced solutions remedy proposed multiband robustness model following wellestablished methodology robust optimization thus produced robust solutions protected deviations input traffic data general already constitutes challenging problem even commercial solvers like cplex accounting robustness considering multiple time periods effect increasing complexity problem matter fact solutions found cplex low quality associated large optimality gaps overcome difficulties defined hybrid primal heuristic based combination randomized fixing algorithm inspired ant table experimental results aco gapar gapsp gapip colony optimization exact large neighborhood search computational experiments set realistic instances sndlib confirmed heuristic drastically outperforms cplex finding high quality solutions associated low optimality gaps short amount time believe many best solutions found actually optimal future objective characterize appropriate families valid inequalities problem attempt close gaps thus possibly prove optimality found solutions furthermore excellent computational performance suggests possibility using heuristic conveniently adapted applications general settings example paths commodity predetermined expect hybrid primal heuristic perform well even different contexts acknowledgements work partially supported german research foundation dfg project multiperiod network optimization dfg research center matheon project einstein campus mathematics berlin ecmath project rouan german federal ministry education research bmbf project robukom bauschert grant appendix proofs statements section appendix proof proposition statement let dtcp exists optimal solution dtc proof prove statement induction considering two consecutive time periods basis step note relation trivially holds definition dtcp inductive step suppose relation holds show holds also period suppose relation hold commodity corresponding path without loss generality assume exists unique respect relation period argument use applied iteratively number deviating inequalities moreover suppose shortest path edge cost let residual flow simplicity assume routed single path argument similarly holds splits several paths given assumption easy see routed another objective function value path gain another shortest path get equality see gives equivalent optimal solution terms objective function value otherwise contradiction suppose shortest path edge cost argument applies solution gains case reroute period period holds new solution deviating inequality gains cost per unit capacity argument holds appendix proof theorem statement consider problem namely linear relaxation let shortest path edge length commodity optimal solution defined routing time period entire flow commodity shortest path installing edge exact capacity needed route traffic flow dct proof proceeding proof recall capacity variables lose integrality constraint thus capacity installation edge period exactly equal flow well known result stated theorem holds one single period see ahuja propose proof result extending reasoning case multiple periods consider problem case apex thus equal involved quantities commodity suppose variable optimal solution problem hence contribution thepobjective function let shortest path thus theorem holds single period case assume theorem holds periods show holds periods proposition may assume optimal solution periods let denote optimal objective function value periods according induction hypothesis also let denote contribution optimal solution periods objective function period suppose determined commodity shortest path edge cost period using similar argument hence single period case easy see increase demand commodity period routed shortest paths used previous periods case let denote contribution optimal solution objective function period determined commodity suppose shortest path edge cost period clear hence references references ahuja magnanti orlin network flows theory algorithms applications prentice hall upper saddle river bauschert andreagiovanni koster kutschka steglich network planning demand uncertainty robust optimization ieee communications magazine doi belotti kompella noronha comparison otn mpls networks traffific uncertainty submitted http retrieved ghaoui nemirovski robust optimization springer heidelberg bertsekas network optimization continuous discrete models athena scientific belmont bertsimas brown caramanis theory applications robust optimization siam review bertsimas sim price robustness oper res bienstock chopra tsai minimum cost capacity installation multicommodity network flows math prog bley andreagiovanni hanemann robustness communication networks scenarios mathematical approaches proc itg symposium photonic networks vde verlag berlin bley andreagiovanni karch wdm fiber replacement scheduling proc inoc electronic notes discrete mathematics doi bley design broadband virtual private networks model heuristic dimacs series discrete mathematics theoretical computer science blum ant colony optimization introduction recent trends physics life rew puchinger raidl hybrid metaheuristics combinatorial optimization survey appl soft comp andreagiovanni new results uncertainty robust optimization klasing experimental algorithms sea lncs vol springer heidelberg doi andreagiovanni robust optimization uncertainty part theory corr http andreagiovanni new theoretical framework robust optimization uncertainty helber eds operations research proceedings springer heidelberg andreagiovanni raymond multiband robust optimization huisman eds operations research proceedings springer heidelberg christofides brooker optimal expansion existing network math program crainic gendreau cooperative parallel tabu search capacitated network design heuristics danna rothberg pape exploring relaxation induced neighborhoods improve mip solutions math program andreagiovanni krolikowski pulaj hybrid primal heuristic robust multiperiod network design appear applications evolutionary computation european conference evoapplications lecture notes computer sciences springer heidelberg dely andreagiovanni kassler fair optimization meshconnected wlan hotspots wireless communications mobile computing doi dorigo caro gambardella ant algorithms discrete optimization artificial life dorigo maniezzo colorni ant system optimization colony cooperating agents ieee trans syst man cybern gambardella montemanni weyland coupling ant colony systems strong local searches europ oper res gamvros raghavan traffic routing satellite networks europ oper res gendreau potvin smires soriano capacity expansion local access telecommunications network europ oper gupta grossmann efficient multiperiod minlp model optimal planning offshore oil gas field infrastructure ind eng chem ibm ilog cplex http kleeman seibert lamont hopkinson graham solving multicommodity capacitated network design problems using multiobjective evolutionary algorithms ieee trans evol comp koster kutschka raack robust network design formulations valid inequalities computations networks lardeux nace geffard multiperiod network design incremental routing networks maniezzo exact approximate nondeterministic procedures quadratic assignment problem informs comp orlowski pioro tomaszewski sndlib survivable network design library networks theimer towards scalable flat core network ecoc vienna austria retrieved http thomas
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feb problem existence conjugacy injectors generalized xia yin nanying yang school science jiangnan university wuxi china yangny vorob department mathematics masherov vitebsk state university vitebsk belarus vorobyovnt abstract paper prove existence conjugacy injectors generalized groups hartley class defined invariable hartley function give description structure injectors introduction throughout paper groups finite prime always denotes group order set primes dividing denotes set primes let set primes let use denote hall recall class groups called fitting class closed taking normal subgroups products normal usual denote classes groups soluble groups nilpotent groups respectively denote classes soluble nilpotent respectively denote classes groups groups respectively well known classes fitting classes definition fitting class see fitting class every group unique maximal normal called denoted class groups subgroup said whenever subgroup group said research supported nnsf grant china grant natural science foundation jiangsu province grant keywords finite group fitting class injector group mathematics subject classification subgroup every subnormal subgroup clearly every subgroup fitting classes play important role theory groups importance theory fitting classes firstly seen following theorem proved fischer harley fact generalization classical sylow theorem hall theorem theorem see theorem let fitting class soluble group possesses exactly one conjugacy class fitting class let note fitting class group sylow hall group hall see existence shemetkov posed following problem problem shemetkov problem let fitting class soluble groups true every finite group possesses problem resolved fitting classes connection interesting problem find conjugate class injectors groups first result direction following famous chuchin theorem possesses hall two hall conjugate development chuchin theorem shemetkov guo proved following theorem theorem let fitting class two conjugate product two fitting classes class well known product two fitting classes also fitting class multiplication fitting classes satisfies associative law see theorem following function nonempty fitting classes called hartley function brevity fitting class local let supp called support fitting class called hartley class exists case said defined clear hence hartley class local fitting class see converse true general see two class functions write concerning fitting classes injectors authors see also posed problem universe let local fitting class soluble groups could describe soluble group view problem theorem following general question class naturally arise problem local fitting class group particular group whether possesses two conjugate note exist groups fitting classes see example paper developing local method offered hartley resolve problem partial groups particular groups hartley class defined invariable hartley function fact prove following theorem support hartley class defined invariable hartley function prime supp fitting class particular group following statements hold possesses two conjugate every type subgroup hall theorem series new classical conjugate classes injectors group obtained structure injectors groups described example following results directly follow theorem corollary every group possesses exactly one conjugacy class corollary every group injector limited length natural number two conjugate theorem following corollary hartley class type nonempty fitting class group possesses two conjugate case class identity groups taking account theorem corollary every group possesses exactly one conjugacy class injectors injector maximal subgroup containing pnilpotent radical note even statements theorem corollaries still new results soluble group unexplained notations terminologies standard reader referred necessary preliminaries recall said exists series subnormal subgroups every factor either particular group said said subgroup said hall said normal hall said said note resp fitting subgroup subgroup resp called nilpotent radical radical resp denoted maximal normal maximal normal said denoted denoted lemma see theorems corollary support particular following results well known see example remarks lemma let class groups isomorphism subgroup maximal normal subgroup subgroup lemma lemma theorem let fitting class subnormal subgroup fitting class gfh lemma see chapter theorem every contained two conjugate definition let supp support hartley class said integrated full different primes full integrated full integrated well invariable easy see every hartley class defined integrated invariable full integrated fact since proof theorem proof theorem consists large number steps following lemmas main steps lemma every group possesses exactly one conjugacy class injectors injector product hall proof prove lemma induction let maximal normal subgroup consider following two possible cases case identity group induction possesses exactly one conjugacy class injectors every injector hall let hall since every hall soluble two conjugate since lemma view conjugacy injectors may assume since hall group nilpotent every injector hence prove maximal subgroup injector lemma support maximal subgroup since normal hence since lemma means maximal subgroup shows statement lemma holds case case let lemma hence case possesses exactly one conjugate class injectors type hall set injectors coincide set hall since soluble theorem say lemma subgroup follows subgroup prove maximal subgroup assume maximal subgroup hall without loss generality may assume hence since follows lemma hence shows thus maximal subgroup order prove injector lemma need prove injector induction injector type mge hall since two conjugate theorem may without loss generality assume since injector maximal subgroup therefore shows injector therefore existence injector group proved conjugacy injectors follows cojugacy hall group shows lemma also holds case lemma proved case lemma directly obtain following corollary every group possesses exactly one conjugate class injectors every injector type sylow support set groups use fitx denotes fitting class generated fitx smallest fitting class containing class groups use denotes lemma every hartley class defined full integrated different primes supp proof let hartley class integrated following define let group group since every fitting class closed respect normal subgroup shows ing equation moreover since therefore obtain let fit let prove fact since equation see fit fit therefore fit fit shows thus obtain moreover since integrated integrated order complete proof lemma need prove different primes fact let clearly lnq however since hence lnq follows lnq shows every group contained therefore group thereby induces thus fit fit completes proof lemma lemma hartley class may always assume defined full integrated call subgroup lemma let full integrated group particular subgroup containing belongs proof assume implies first prove let since hence consequently hand since isomorphism follows order prove need prove need prove since lemma let arbitrary prime since since full hence shows different primes hence isomorphism thus since hypothesis clearly hence consequently see implies conversely similar argument proof see completes proof lemma let full integrated supp group injector proof prove lemma induction let maximal normal subgroup different primes first prove primes fact since different primes isomorphism implies let since primes hence follows thus consider following two possible cases case since lemma since class groups fitting class theorem also hence induction assume subgroup since subgroup hence maximal subgroup follows lemma lemma maximal subgroup contradicts thus subgroup hence lemma case since injector lemma induction similar argument case obtain lemma proved view lemmas may prove main theorem proof theorem let maximal normal subgroup note invariable hence full integrated since therefore injector lemma clearly arguments proof lemma obtain prove injector fact assume subgroup subnormal subgroup order prove injector need prove subgroup every subnormal subgroup since assume subgroup let subgroup clearly subgroup contradiction shows injector assume another injector hence lemma conjugate implies conjugate hence let injector hence lemma hall lemma shows thus holds theorem proved lemma known every group possesses exactly one conjugacy class injectors note also every group connection put forward following question question support group true possesses exactly one conjugacy class injectors references fischer hartley injektoren endlicher gruppen math doerk hawkes finite soluble groups walter gruyter berlin new york guo theory classes groups science academic publishers beijingnew ezquerro classes finite groups springer dordrecht mazurov khuhro kourovka notebook unsolved problems group theory institute mathematics siberian branch ras novosibirsk nilpotent injector finite groups bull austral math iranzo existencia injectores groups finites respecto ciertas class fitting publ mat univ autonima baselona existence conjugacy subgroups finite group mat sbornik shemetkov subgroups groups finite groups hauka tehnika minsk guo injectors finite groups chinese ann math ser hartley fischer dualization formation theory proc london math vorob hawkes conjecture radical classes siberian math hauck zahurski characterization dominant local fitting classes algebra liu guo vorob description finite soluble groups math sci rec iranzo toress finite group rend sem mat uni padova guo structure theory canonical classes finite groups springer shemetkov formations finite groups moscow nauka main editorial board physical mathematical literature suzuki group theory new york iranzo monazor respect fitting class arch
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sublinear time estimation degree distribution moments degeneracy connection full version feb talya dana abstract revisit classic problem estimating degree distribution moments undirected graph consider anp undirected graph vertices define dsv aim estimate within multiplicative error given approximation parameter sublinear time consider sparse graph model allows access uniform random vertices queries degree vertex queries neighbor vertex case average queries suffice constant feige sicomp degree rsa sidma extended result integral queries strictly speaking algodesigning algorithms performs rithm approximates number given size slight modification gives algorithm moments design new significantly simpler algorithm problem exactly matches bounds much simpler proof importantly running time algorithm connected degeneracy essentially maximum density induced subgraph family degeneracy query complexity min thus class bounded degeneracy graphs includes minor closed families preferential attachment graphs estimate average degree queries estimate variance degree distribution queries major improvement previous bounds key insight designing estimator low variance large dense subgraphs introduction estimating mean moments sequence integers classic problem statistics requires little introduction absence knowledge moments sequence possible prove anything suppose integers formed degree sequence graph formally let undirected graph vertices school tel aviv university research partially supported grant blavatnik fund author grateful azrieli foundation award azrieli fellowship school tel aviv university danaron research partially supported israel science foundation grant grant blavatnik fund department computer science university california santa cruz sesh let denote degree vertex assume every feige proved uniform random vertex degrees expectation suffice provide approximation average degree use suppress poly log factors variance large graphs constant average degree simply consider star constraints degree distribution allow approximations classic theorems characterize sequences star graph shows beaten sublinear time pure vertex sampling suppose could also access random neighbors given vertex setting goldreich ron showed possible obtain average degree expected time substantial complex generalization gonen ron shavitt henceforth grs gave algorithm estimates higher moments degree distribution technically grs gave algorithm approximating number stars graph simple modification yields algorithm moments estimation precision let formally define problem degree distribution distribution pdegree uniform random vertex moment degree distribution dsv degree distribution moment estimation ddme problem let graph vertices known access provided following queries get label uniform random vertex query degree vertex iii query uniform random neighbor vertex given output approximation ddme problem important connections network science study properties graphs numerous results significance degree distributions graphs since seminal results degree distribution moments commonly used characterize model graphs appearing varied applications theoretical side recent results provide faster algorithms graphs degree distribution specified form practical algorithms specific cases ddme studied dasgupta chierichetti results requires bounds mixing time random walk results let denote number edges graph provided algorithm sake simplicity restrict discussion introduction case observed grs ddme problem smaller significantly larger algorithm ddme grs designed expected proved expression optimal poly log dependencies also suppresses additional factors depend note graph without isolated vertices every yields bound case estimating average degree recovers bounds mention recent result aliakbarpour ddme stronger model assumes additional access uniform random edges get better bound stronger model note main challenge ddme assumption isolated vertices made sake simplicity presentation ensures basic lower bound moments constant matter convenience increased least taking median value log independent invocations measuring contribution vertices becomes substantially easier random edges provided ddme problem without samples quite even detect high degree vertices bounds given known optimal poly log dependencies first blush problem appears solved unearth connection ddme degeneracy degeneracy factor maximum density subgraphs design algorithm nuanced query complexity depending degeneracy result subsumes existing results provides substantial improvements many interesting cases furthermore algorithm analysis significantly simpler concise grs result begin convenient corollary main theorem tighter precise bound appears theorem theorem consider family graphs degeneracy ddme problem solved family using min queries expectation running time linear number queries consider case bounded degeneracy graphs rich class graphs every family graphs bounded degeneracy graphs generated preferential attachment process rich theory bounded expansion graphs spans logic graph minor theory tractability graph classes bounded degeneracy every class graphs get time stress bounded degeneracy imply bounds maximum degree moments star graph degeneracy extremely large moments due central vertex consider bounded degeneracy graph without isolated vertices accurately estimate average degree poly log queries estimate variance degree distribution poly log queries contrast timal bounds feige average degree bound grs variance estimation general bound significant improvement bound grs algorithm attaining theorem requires upper bound degeneracy graph degeneracy bound given algorithm recovers bounds grs improvement extra poly log factors details theorem note bound theorem attained algorithm given parameter particular algorithm provided must work graphs vertices must perform queries order approximate average degree even graphs constant degeneracy constant average degree details given subsection full version paper bound theorem may appear artificial prove optimal general case also optimal upper lower bounds construction extension lower bound proof grs theorem consider family graphs algorithm ddme problem family requires min queries degeneracy moment estimation begin closer look lower bound examples feige grs core idea quite simple ddme hard overall graph sparse small dense subgraphs consider case clique size connected tree size small clique dominates average degree sublinear algorithm access random vertices pays approximation grs use complex constructions get lower bound general also involves embedding small dense subgraphs dominate moments prove converse lower bound constructions words prove dense subgraphs must imply ddme easier convenient parameter degeneracy degeneracy global parameter clear sublinear algorithm exploit furthermore ddme algorithms typically local sample random vertices query degrees vertices maybe also query degrees neighbors need local property sublinear algorithms exploit also linked degeneracy achieve connection via degree ordering consider dag obtained directing edges lower higher degree vertices related properties distribution degeneracy exploited clique counting nonetheless clear link ddme use techniques state result merely show led use degree ordering main insight construction estimator ddme whose variance depends degeneracy estimator critically uses degree ordering proof relates variance estimator density subgraphs bounded degeneracy stress algorithm quite simple technicalities analysis setting certain parameters designing algorithm designate weight edge simple calculation yields sum weights edges exactly suppose could sample uniform random edges knew total number edges could hope estimate uniform edge sampling variance edge weights bounded yields algorithm vertex isolated indeed similar approach aliakbarpour variance calculations also used classic result frequency moment estimation approach simulate uniform edge samples using uniform vertex samples suppose sampled set uniform random vertices querying degrees vertices select vertices probability proportional degrees allows uniformly sample edges incident vertices simply run uniform edge sampling algorithm edges algorithmic structure recently used sublinear triangle counting algorithms eden lies core technical challenge bound number random vertices sufficient effectively simulating random edge algorithm boils behavior variance vertex weight distribution let weight vertex sum weights incident edges weight distribution vertices extremely skewed approach would require forbiddingly large standard technique triangle counting first introduced helps reduce variance direct edges lower degree higher degree vertices breaking ties consistently set weight vertex sum weights incident edges thus vertex lower degree neighbors significantly reduced weight reducing overall variance general case ignoring degeneracy relatively simple argument bounds maximum weight vertex enables bound variance weight distribution yields much simpler algorithm proof grs bound case graphs bounded degeneracy need refined approach key insight intimate connection variance existence dense subgraphs basically show main structure leads high variance existence dense subgraphs formally translate small upper bound density subgraph bound variance vertex weights establishes connection graph degenearcy simplicity algorithm viewpoint ddme quite different grs precursor proceed bucketing vertices based degree leads complicated algorithm essentially samples estimate size buckets also number edges various buckets make use buckets analysis order obtain upper bound depends degeneracy order achieve grs upper bound analysis use bucketing explained main ddme procedure simple enough present lines pseudocode see figure feel structural simplicity important contribution work takes two sampling parameters main result theorem follows running standard geometric search right setting use denote label vertex vertices unique ids complete order ids select vertices uniformly independently random let resulting denoted query degree vertex let select vertex probability proportional degree probability dvi query random neighbor else set dvi dui dvi dui set return dqr figure algorithm approximating related work mentioned beginning section aliakbarpour consider problem approximating number given access uniformly selected edges given ability uniformly select edges select vertices probability proportional degree rather uniformly used get unbiased estimator count low variance leads bound optimal dasgupta kumar sarlos give practical algorithms average degree estimation though assume bounds mixing time random walk graph recent paper chierichetti build methods sample nodes according powers degree closely related ddme simpson seshadhri mcgregor give practical algorithms estimate entire cumulative degree distribution streaming setting different sublinear query model consider results mostly empirical eden present algorithm approximating number triangles graph although different problem ddme similar challenges regarding vertices indeed mentioned earlier approach sampling random edges set random vertices used degeneracy closely related density notions arboricity thickness strength graph rich history algorithmic results run time depends degeneracy sublinear algorithms estimating various graph parameters include approximating size spanning tree maximum matching minimum vertex cover main theorem theorem every graph exists algorithm returns value probability least assume algorithm given upper bound degeneracy bound provided algorithm assumes trivial bound expected running time minimum following two expressions log log log min log log log min equation essentially query complexity grs albeit better dependence log thus algorithm guaranteed least good exactly degeneracy prove equation less equation within expression min two terms first term smaller iff mechanism deriving rather cumbersome running time following algorithm theorem runs geometrically increasing values turn derived geometrically decreasing guess uses guess set setting values depending setting independent algorithm simply picks minimum settings achieve smaller running time sufficient conditions section provide sufficient conditions parameters used figure order algorithm return estimate first introduce notations graph vertex let denote set neighbors set vertices let set edges incident vertices think edges ordered pairs thus distinct observe defined step equals let analysis algorithm convenient work instead critical aspect algorithm proof degree ordering vertices formally set given degree ordering let elsewhere use shorthand definition define weight edge follows define otherwise set vertex vertices observe given notations definition selects uniform edges sets step based definition obtain next two claims first claim connects weights vertices claim proof definition weights dsv claim established claim exp defined step algorithm proof recall note defined step algorithm exactly conditioning exp definition algorithm see step exp exp let remove conditioning since linearity expectation expr thus using claim exp expr exp completes proof conditions parameters next state two conditions parameters used algorithm establish several claims based conditions holding conditions stated terms properties graph well approximation parameter confidence parameter vertex condition edge condition lemma condition holds probability least following hold proof first look random variable definition claim exp turning variance since vertices chosen uniformly random varr varv chebyshev inequality exp exp exp applying lower bound condition probability indeed condition defined get bound bounds follow simply markov inequality observe exp random variable satisfies exp markov inequality apply union bound complete proof theorem conditions hold probability least proof condition choice exp turning variance since edges chosen uniformly random var var var exp let condition bounds lemma hold note chosen probability least get var apply chebyshev inequality invoke condition exp var lemma exp taking account probability satisfy one bounds lemma probability conditioned satisfying bounds deviates expected value get probability least algorithm edge samples aside analysis slightly adapted prove result aliakbarpour estimating moments using random edge queries observe simply set immediately gives shown equation similarly shown proof theorem var defined step algorithm shown equation thus set sufficiently large constant satisfy condition exactly bound aliakbarpour satisfying conditions general graphs show set satisfy conditions general graphs setting give query complexity dependence log improve exponential dependence incur next section show setting improved using degeneracy bound sufficiently large constants set min setting parameters requires knowledge exactly trying approximate normalization factor simple geometric search argument alleviates need know details see section follows elsewhere make use inequality theorem inequality values refer conjugates formula order assert set equation satisfies condition suffices establish next lemma lemma condition holds proof let degree threshold define partition vertices vertices puseful upper bounding maximum weight vertex hence upper bounding details follow first observe true since otherwise dsv contradiction claim upper bound implies max verify assume contrary claim contradicts bound least vertices also useful bound inequality conjugates bound dsu mss turn bounding maxv definition degree ordering first term equation recall thus equation second term using applying equation finally max second inequality follows combining equations get upper bound maxv applying claim next lemma implies condition holds set equation lemma condition holds min proof bound two ways first standard norm inequality since dsv also use trivial bound get thus min applying inequality conjugates get multiply bound complete proof degeneracy connection degeneracy coloring number graph maximum value subgraphs minimum degree definition replace minimum average get approximation degeneracy refer theorem corollary abusing notation convenient define also make following observation regarding relation claim every graph proof let subset vertices maximizes let denote average degree subgraph induced hence inequality theorem conjugates dsv implying since get section show following setting parameters satisfies conditions every graph degeneracy appropriate constants min min clearly setting equation equation respectively improve setting general case equation section equation main challenge proving condition holds set graph degeneracy goal upper bound however opposed proof lemma section simply obtained upper bound maxv bounded maxv analysis refined uses degeneracy bound details see proof main lemma stated next lemma condition holds sufficiently large constant order prove lemma first introduce following definitions claim definition set vertex let denote set let denote set two sets vertices necessarily disjoint let definition partition vertices degree least according degree let partition vertices according number outgoing edges specifically define also define hence partition central building block proof lemma next claim claim establishes upper bound number edges going vertices vertices every within appropriate intervals proof claim exploit degeneracy bound claim let defined definition let defined definition every every proof since degeneracy hand definition combining two bounds get obtain following bound number edges neighbors next upper bound definition every exists vertex definition every implying every every holds hence every also definition every holds therefore dsu directly implies proof follows plugging equation equation ready prove lemma proof lemma definition since every order bound expression equation consider defined definition separately applying inequality conjugates get following bound every dsu vertex let applying equation one term square equation follows definition switching order summations equation follows splitting sum equation based partition subsets recall fixed defined definition point consider empty definition trivially upper bound thus dsu turning since vertices degree equation get hence claim using bound equation bound equation plugging equation get log last inequality holds fact save log factor using inequality equation implies lemma follows combining equation equation remains establish condition lemma condition holds min proof since bounded degeneracy follows equation shown proof lemma min proof follows two bounds case estimating average degree average degree simple analysis slight variant observe definition every edge degree threshold let definition since graph degeneracy implies dvi suppose modify algorithm set well otherwise set modification exp since get therefore thus compared setting order satisfy condition suffices set equation save log factor cost factor importantly analysis simple compared proof lemma setting equation gives wrapping things proof final result theorem follows combining theorem lemma lemma lemma geometric search estimate determines correct setting algorithm convenience restate bounds theorem terms log log log min log log log min proof proof theorem recall setting equation equation respectively equals setting equation set maximum possible value hence suffices prove theorem assumption algorithm provided upper bounds follows theorem lemma lemma lemma invoked parameters set equation equation respectively algorithm returns value probability least however settings require knowledge parameter trying approximate normalization factor hence use following search algorithm maximum possible value compute start guess according equation according equation assuming given approximation parameter invoke stop log log times let median returned values return otherwise halve repeat observe decreasing functions hence running time invocation running time last invocation claim markov inequality invocation value returned algorithm stress holds regardless whether satisfy conditions respectively definition values probability stop step log therefore high constant probability stop satisfy conditions theorem invocation probability least probability least log thus algorithm stop return value probability least log union bound iterations algorithm returns value high constant probability order bound expected running time first observe running time times invocation hand probability running time invocation log algorithm stop reach bound expected running time follows lower bounds bounded degeneracy lower bounds given section hold algorithms allowed degree neighbor queries well pair queries queries form edge lower bound show complexity algorithm stated theorem graphs degeneracy tight dependence polylogarithmic factors constant even log log theorem approximation algorithm queries must perform next define two families proof every graphs graph consists clique vertices independent set vertices graph vertices partitioned three sets sizes respectively set clique connected complete bipartite graph edges within set independent set within family graphs differ invoke precise order satisfy conditions values recall claim note dsv labeling vertices construction families graphs degeneracy clearly unless algorithm hits vertex clique graph belonging vertex graph belonging distinguish graph selected randomly graph selected randomly probability hitting thus order event occur high constant probability vertex queries necessary theorem approximation algorithm must perform min queries proof proof theorem based simple modifications lower bound constructions thm note theorem stated explicitly algorithms perform degree neighbor queries since algorithm presented well current paper however noted sec also hold pair queries allowed essentially based hitting special vertices theorem proved considering two cases defined according relation namely constant determines hardness approximation within case two defined according relation namely four gives one terms lower bound within min expression consider correspond construct two families graphs values families graphs degeneracy every graph satisfies lower bound based every graph satisfies difficulty distinguishing random graph selected random graph selected specified number queries cases modify construction thm either decreasing degeneracy increasing case modify construction described item case thm families vertices partitioned two subsets size graph set independent set graph vertex neighbors appropriate setting families vertices degree appropriate setting neighbors neighbors observe graphs family viewed obtained graphs replacing pairs edges single edge refer edges graphs edges replacing graphs special edges settings number stars graphs belonging roughly factor larger number stars graphs belonging difficulty distinguishing random graph selected random graph selected based upper bounding following two similar events hitting vertex querying graph either performing query performing neighbor query receiving answer hitting vertex special edge vertices querying graph number special edges follows modify settings defined item thm case large perform additional small modification set ensures graph set hence graphs families graphs since number edges graphs belonging order number special edges graphs belong get lower total number edges course requires formalizing done bound set still holds graphs families within add edges form clique subset size thus increasing degeneracy since modification families effect ability distinguish two families since number edges including clique get lower bound case may assume without loss generality case sufficiently large constant else lower bound trivial may trivial similarly else lower bound also assume since every graph claim modify construction described item thm construction except size set needs similar one described graphs vertex increased specifically let connected every vertex graph therefore families vertex degree since due sufficiently large constant get graphs difference families graphs belonging vertex neighbors neighbors graphs belonging vertex neighbors vertex neighbors vertex implies used since number edges graph number corresponding special edges within graph total number edges get lower bound use construction plant clique size degeneracy hence increased due clique since number value increased get lower bound edges including clique knowing degeneracy bound one may wonder query complexity theorem obtained without knowledge degeneracy ideal situation would one algorithm complexity without knowing lower bound preclude possibility since uses graphs high degeneracy nonetheless slight adaptation arguments shows bound holds even bounded degeneracy graphs algorithm must work graphs focus solely estimating average degree since suffices make point definition constant algorithm called given query access graph probability outputs average degree note access degeneracy bound valid required accurate graphs given degeneracy bound graphs theorem let sufficiently large integer consider class graphs vertices degeneracy constant algorithm must perform queries graphs proof similarly previous proofs sufficiently large define two distributions labeled graphs consider graph consisting two connected components cycle vertices cycle vertices constant statement theorem distribution generated labeling vertices using uniform random permutation second distribution take graph consists cycle vertices clique vertices distribution generated labeling vertices according uniform random permutation consider possibly randomized algorithm deciding given query access either graph generated graph generated according distribution graph generated long algorithm perform query vertex belongs small cycle graph generated small clique graph generated answers queries identically distributed two distribution implies decision algorithm must perform queries order succeed probability least consider algorithm makes queries graph vertices degeneracy use algorithm order construct algorithm distinguishes queries works follows given query access runs independent runs run makes queries terminates outputs end runs provided estimate average degree median outputs otherwise outputs suppose recall average degree exactly runs guaranteed make queries thus output estimate validity chernoff bound median estimate probability least gives correct output suppose recall average degree least run takes queries outputs correctly otherwise similar argument one made median estimate least probability least thus correct query complexity distinguishes probability least therefore must references arboricity wikipedia https graph degeneracy wikipedia https aliakbarpour biswas gouleakis peebles rubinfeld yodpinyanee algorithms counting star subgraphs applications join selectivity estimation 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journal computing goldreich ron approximating average parameters graphs random structures algorithms gonen ron shavitt counting stars small subgraphs sublineartime siam journal discrete math hakimi realizability set integers degrees vertices graph siam applied math hassidim kelner nguyen onak local graph partitions approximation testing proceedings fiftieth annual symposium foundations computer science focs pages ieee havel remark existence finite graphs czech casopis pest marko ron approximating distance properties general sparse graphs acm transactions algorithms matula beck ordering clustering graph coloring algorithms jacm ossana mendez sparsity springer nguyen onak approximation algorithms via local improvements proceedings annual symposium foundations computer science focs pages ieee onak ron rosen rubinfeld algorithm approximating minimum vertex cover size proceedings annual symposium discrete algorithms pages siam parnas ron approximating minimum vertex cover sublinear time 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jun marginal estimation parameter driven binomial time series models dunsmuir school mathematics statistics university new south wales sydney australia school mathematics statistics university new south wales sydney australia abstract paper develops asymptotic theory estimation parameters regression models binomial response time series serial dependence present latent process use generalized linear model glm estimating equations leads asymptotically biased estimates regression coefficients binomial responses alternative use marginal likelihood variance latent process serial dependence accounted practice equivalent using generalized linear mixed model estimation procedures treating observations independent random effect intercept term regression model prove method leads consistent asymptotically normal estimates even autocorrelated latent process simulations suggest use marginal likelihood lead glm estimates result problem reduces rapidly increasing number binomial trials time point binary data chance remain even long time series provide combination theoretical heuristic explanations phenomenon terms properties regression component model used guide application method practice keywords phrases binomial time series regression parameter driven models marginal likelihood introduction discrete valued time series increasingly practical importance applications diverse fields analysis crime statistics econometric modelling high frequency financial data animal behaviour epidemiological assessments disease outbreak monitoring modern biology including dna sequence analysis see dunsmuir tran weatherburn paper focus time series binomial counts two broad classes models time series counts based categorization cox generally discussed literature observation driven models serial dependence relies previous observations residuals parameter driven models serial dependence introduced unobserved latent process estimation parameter driven models significantly challenging especially latent process correlated therefore methods provides preliminary information regression parameters without requiring heavy computation load would appealing example use generalized linear model glm estimation obtaining estimates regression parameters discussed davis dunsmuir wang davis poisson negative binomial observations respectively glm estimation consistent asymptotically normal two types response distribution even latent process inducing serial dependence however recently pointed cui discussed detail use glm binary binomial data leads asymptotically biased estimates cui propose semiparametric estimation method binary response data marginal probability success modelled paper takes different approach suggest using estimation based marginal distributions accounts variance latent process serial dependence procedure easy implement using standard software fitting generalized linear mixed models glmm show method leads estimates regression parameters variance latent process consistent asymptotically normal even latent process includes serial dependence additionally method extends easily response distributions poisson negative binomial estimation binomial time series mixed models cases improve efficiency regression parameters related glm estimates suppose represents number successes trials observed time assume observations process xnt observed vector regressors may depend sample size form triangular array whose first component unity intercept term given xnt latent process independent density fyt exp xtnt log exp log var process observed referred latent process often assumed stationary gaussian linear process zero mean cov marginal variance parameters serial dependence model specification stationary gaussian linear process covers many practical applications assume remainder paper however gaussianity required main asymptotic results presented general assumed stationary strongly mixing process discuss extension section let denote collection parameters let true parameter vector model likelihood defined terms integral dimension follows exp joint density given parameters maximization likelihood computationally expensive methods estimating high dimensional integrals using approximations monte carlo method reviewed davis dunsmuir however simple implement methods provide asymptotically normal unbiased estimators without need fit full likelihood useful construction statistics needed investigate strength form serial dependence also provide initial parameter values maximization full likelihood practitioners glm estimation strong appeal easy fit standard software packages glm estimators regression parameters obtained treating observations independent xtnt using glm log xtnt xtnt let denote value maximises glm estimate assumes additional unexplained variation responses beyond due regressors xnt however recently noted cui glm provide consistent estimates contains latent autocorrelated component specific deterministic regressors xnt example limit estimation binomial time series mixed models show converges maximizes equivalently unique vector solves poisson negative binomial cases xtnt xtnt modifies regression intercept influence response regression terms identity usually hold binomial observations relationship binomial logit regression models investigated several researchers example neuhaus kalbfleisch hauck proved logit marginal probability approximated single covariate equality attained wang louis proved bridge distribution logit equals holds cui proposed mglm method glm estimates binomial observations generated model inconsistent overcome inconsistency observed glm estimation paper propose use marginal likelihood estimation maximises likelihood constructed assumptions process consists independent identically distributed random variables assumption full likelihood replaced marginal likelihood exp corresponding marginal function log exp log density mean zero variance normal random variable let estimates obtained maximising compact parameter space marginal likelihood estimators easily obtained standard software packages fitting generalized linear mixed models since marginal likelihood estimates consistent used starting value full likelihood based additionally asymptotic distribution standard deviation derived asymptotic covariance matrix used assess significance regression parameters moreover another paper developed test first detect existence latent process present whether serial dependence asymptotic results paper needed order derive large sample distribution second step score test detecting serial dependence large sample properties marginal likelihood estimates provided section simulations section show marginal likelihood estimates lead high probability number trials small particular almost binary data hence section focuses obtaining asymptotic approximations upper bound useful quantify proportion times marginal likelihood procedures degenerates glm procedure also section derive theoretical mixture distribution provided better approximation situation section presents simulation evidence demonstrate accuracy asymptotic theory covariance matrix marginal likelihood estimate section discusses difference marginal likelihood estimation mglm estimation cui section concludes asymptotic theory marginal likelihood estimates present large sample properties marginal likelihood estimates obtained maximizing begin presenting required conditions latent process regressors xnt sequence binomial trials estimation binomial time series mixed models process strongly mixing sup sup generated respectively practice number trials may vary time allow introduce condition sequence trials stationary strongly mixing process independent mixing coefficients satisfy let assume alternative would take deterministic asymptotically stationary case would limits finite sample frequencies occurrences specifications obviously include case yields binary responses previous literature davis dunsmuir wang davis cui allow deterministic stochastic regressors condition regression sequence specified one two ways deterministic covariates defined functions xnt specified piecewise continuous vector function stochastic covariates stationary vector process xnt observed trajectory stationary process condition let dim regressor space xnt assume rank span full rank space spanned regressors required condition holds many examples instance deterministic regressors generated functions given condition exist different values corresponding function linearly independent stochastic regressors generated stationary process given condition linearly independent found almost surely cov condition latent process strictly stationary gaussian strongly mixing mixing coefficients satisfying conditions unique asymptotic limit marginal likelihood estimators also required denote marginal probability successes trials time ejwt dzt xtnt unit variance gaussian process density function similarly let marginal probability evaluated true values time define conditional xnt log conditions let evaluated condition log estimation binomial time series mixed models condition log proof included proof theorem condition unique maximum true value establish consistency asymptotic normality marginal likelihood estimator theorem consistency asymptotic normality marginal likelihood estimators assume conditions lim cov lim log xnt xnt dzt use theorem practice requires least identifiability condition holds covariance estimated single series address aspects detail section addition particularly binary responses marginal likelihood estimators produce high probability address detail section propose improved asymptotic distribution based mixture asymptotic identifiability nowpdiscuss circumstances pmwhich condition holds since log log thus model identifiable lemma assume condition condition holds marginal likelihood proof outlined appendix binary data hence model identifiable everywhere distinct value found establish exi exi exi exi implies xit xit condition linearly independent found establish unique solution hence holds condition holds unique solution xit xit found assume regressor space set discrete vectors let matrix holds exists solution since excluded possible solutions unique solution exists hence exists establishes therefore identifiable note rank overdetermined thus solution always exist situations condition holds however general proof without conditions regressors difficult instead provide rigourous proof show condition holds binary data regressor space connected estimation binomial time series mixed models lemma let addition condition assumed connected subspace condition holds proof see appendix estimation covariance matrix use theorem asymptotic covariance matrix needs estimated using single observed time series estimated replacing marginal likelihood estimates however estimation challenging cross terms estimated without knowledge use modified subsampling methods reviewed cui estimate let denote subseries length starting ith observation total number subseries define replacing similar conditions given show consistent estimator asymptotic covariance matrix performance subsampling estimators relies large extent following guidance heagerty lumley optimal selection use simulations one dimensional integrals easily obtained using function integrate degeneration marginal likelihood estimates even identifiability conditions satisfied finite samples marginal likelihood maximised case degenerates ordinary glm estimate simulation evidence section suggests chance occurring even moderate large sample sizes large binary data decreasing rapidly number trials increases section derive two approximations probability approximations conclude high whenever range xtnt xtnt xtnt constants linear approximation accurate covariance matrix marginal likelihood estimates obtained inverse near singular matrix results var large close nontrivial probability distribution finite samples better approximated mixture two multivariate distributions weighted estimating probability one approximation probability using asymptotic normal distribution provided theorem define limit standard normal distribution alternative approximation based score function evaluated consider scaled score function using integration parts xtnt xtnt estimation binomial time series mixed models implies converse hence bounded order derive large sample approximation probability show section large sample distribution standard normal lim lim var define approximated ncs quantities expressed analytically regression specifications simulations limits computed using numerical integration binary case particular ratio quite small resulting large value compare well estimated via simulations section conclude slightly accurate situation covered asymptotic theory glm estimates marginal score develop asymptotic distribution asymptotic normality required theorem asymptotic normality glm estimators conditions estimates imising likelihood satisfies xtnt xnt xtnt lim lim xnt xnt xns xns xnt xtns lim xtnt xnt xtnt proof theorem given appendix relies concavity glm log likelihood respect standard results functional limit theorem used establish result similar way used davis dunsmuir wang davis cui order use theorem practical purposes first needs determined estimation would require knowledge estimation would require neither estimated using glm procedure hence theorem theoretical value based theorem derive large sample distribution score function marginal likelihood evaluated derivatives uniformly bounded hence jst since xtnt xnt definition using taylor expansion xnt follows xtnt jst xnt xtnt xtnt xtnt xnt lim cnt estimation binomial time series mixed models xtnt xnt vector clt follows clt joint vector note note xtnt sequences strongly mixing proposition blais macgibbon roy clt mixing process proposed davidson applied show asymptotically normally distributed mean zero covariance matrix kst given theorem cov lim lim xnt cov xnt theorem assumptions theorem approximate mixture distribution theorem mixture distribution finite samples assume conditions finite samples distribution approximated mixture distribution obtained skew multivariate normal distribution based joint normality given theorem skew normal distribution based joint normality theorem moreover remarks skew normal distribution defined gupta value mixture similar moran theorem results parallel moran theorem based reasoning however moran results independence observations whereas results require serial dependence accounted asymptotic results mixture provides better theoretical description asymptotic distribution marginal likelihood estimates small practice mixture distribution estimated without knowing true values simulations covariance matrix joint distribution approximated based calculate var var simulation results section summarize results several simulation studies illustrate key theoretical results derived well indicate circumstances large case mixture distribution theorem would provide accurate description estimation binomial time series mixed models examples consider simple linear trend latent process assumed chosen maintain var cases true values varies interval simple example provides substantial insights behaviour marginal likelihood estimates well problems arise simplicity example also allows obtain analytical calculations key quantities provide heuristic explanations distribution results arise particularly binary time series simulations reported later number replications marginal likelihood estimates obtained using package frequency package dependent occasionaly case checked using implementation based adaptive gaussian quadrature comparing results sas proc mixed first simulation section focuses binary responses illustrates distribution marginal likelihood estimates kind data converge towards mixture proposed theorem approximated using result theorem good accuracy second experiment section studies finite sample performance binomial cases shows vanishes increases thus distribution multivariate normal developed theorem finally section method estimation covariance matrix proposed section evaluated order implement simulations section first derive theoretical expressions key quantities used define large sample distributions theorems well estimates analytical expressions asymptotic quantities key quantities required implementation explanation simulation results follow limit point glm estimate appearing theorem used obtain approximation asymptotic variance theorem used obtain approximation various quantities defining mixture distribution theorem throughout derivations given case deterministic regressors specified xnt suitably defined vector function condition first component unity order include intercept term also order reduce notational clutter assume number binomial trials time points analytical expressions involve various integrals computed using numerical integration either integrate using grid evaluation mesh interval calculation theorem theorem obtain variance theoretical upper bound require evaluation integrals form cov integral expression approximated using adaptive gaussian quadratic agq nodes two dimensions limit point glm estimation numerically solving system newton raphson iteration limiting value glm estimates estimation binomial time series mixed models quantities needed analytical expression limiting expectation scaled score respect evaluated limiting point lim xtnt xtnt xtnt xtnt xtnt lim note strictly positive make contribution binary responses case term contributing observed simulations general proof case also observed simulations substantially larger result small responses large binary responses recall lim var case latent process similar expression obtained dependent cases however serial dependence appear make much difference chance least simulations section simulation example binary response case expressions evaluated using numerical integration give since observed strictly positive marginal likelihood ncs theorem probability vanishes bounded marginal likelihood estimate vanishing probability however notice binary data hence even largest sample size reported would require sample size reduce clearly substantial estimation binomial time series mixed models implications use marginal likelihood binary data binomial responses term dominates hence even small values chance reduces rapidly conclude subsection heuristic explanation appearing definition well approximated linearly could approximated first element equal rewritten vector written definition note probability success logit response hence ranges reasonably large values may near linear example used simulations ranges interval interval well approximated straight line asymptotic covariance matrix marginal estimates although positive definite asymptotic covariance matrix theorem near singular results overall covariance matrix large elements particular variance large result given also close reason analyzed section calculating various components asympototic covariance marginal estimates keep discussion manageable deterministic regressors assumed generated functions satisfy condition used derivations case summations left limit given integral first dimension corresponds second dimension corresponds binary responses using facts component let note binary cases define conditional distribution appearing linearly approximation well established fixed nearly linear dependent vector consequently becomes near singular large inverse example takes values interval left side approximately straight line hence view discussion eigenvalues binomial data properties left side longer near linear function result nearly singular moderate size estimation binomial time series mixed models quantities needed mixture distribution assess accuracy asymptotic mixture distribution theoretical mean vector covriance distributions required approximately normal distribution theorem glm estimates mean given section covariance matrix calculated using mean evaluated conditinoal mean covariance matrix example provided binary data var obtained empirically using follows xnt finite samples note although binary data almost linearly dependent analyzed last part nontrivial nonsingular allows nonsingular result large difference theoretical empirical covariance observed binary data shown example simulations example binary data independent latent process example considers simplest case independent observations obtained replication independent binary sequence xnt exp generated table reports empirical values empirical proportion empirical proportion also shown empirical mean standard deviation parentheses conditional obtained simulations along theoretical values obtained associated theorem table clearly demonstrates data generating mechanism high proportion replicates proportion decrease rapidly sample size increasing predicted theory theoretical values provide good approximations explained high proportion zero estimates even large sample sizes expected regression structure used simulation note closer probability estimations reverse theoretical property binary data theoretical approximations decrease slowly large sample required attain also clear table empirical mean standard deviation show good agreement theoretical results predicted theorem overall theory derive use mixture distribution quite accurate sample sizes relatively large sample mixture better representation however estimation corresponding distributions mixture requires therefore implemented practice estimation binomial time series mixed models theoretical empirical simulations standard deviation table mixture distribution marginal likelihood estimates binary independent time series example binomial data correlated latent process simulation investigates bias standard deviation marginal estimates range serial dependence given observed standard deviation estimates replications compared given asymptotic covariance matrix theorem also give empirical proportion event using proportion theoretical upper bound approximated defined table summarizes results binomial series empirical values good agreement asymptotic mean standard deviation bias estimates generally improving increasing also observed binomial series theoretically empirically probability decreases quickly increasing either number trials sample size however binary responses large asymptotic standard deviations obtained reason explained section binary data probability close although theoretically converges zero slowly increases explained theorem settings example using agq calculated vary values example hence binary responses ncs across range autocorrelation considered thus large values ncs across range autocorrelations expected regression structure however binomial series instance ncs dominated explains decreases rapidly increasing interestingly binary series increases bias worsens seems somewhat counterintuitive plausible explanation distribution mixture using weights approximately across sample size serial dependence conditional distribution larger variance resulting inflated overall mean relative summary theoretical upper bound close empirical proportion pattern consistent theorem binomial series marginal estimates good bias properties standard deviations explained large sample distribution theorem conclusions severely impacted level direction serial dependence binary series high proportion persistent regardless level serial dependence explained theory presented example estimate covariance matrix subsampling method estimating covariance marginal likelihood estimates limited practical value binary data firstly occurs nearly time method provide estimates covariance matrix theorem secondly high proportion mixture distribution given theorem covariance requires unknown estimated single sequence binomial case subsampling method likely useful range serial dependence table presents simulation results various levels serial correlation table summarizes estimation binomial time series mixed models mean asd mean asd mean asd mean asd mean asd mean asd table marginal likelihood estimates binomial observations various values true values estimation binomial time series mixed models asd asd table subsampling estimates standard deviation glmm estimation estimates standard deviation using subsampling method described section column asd contains asymptotic standard deviation calculated covariance matrix theorem column contains empirical standard deviation table shows subsampling estimates standard deviations magnitude theoretical standard deviations even moderate sample size biased downwards increasingly increases values provide least biased estimates standard errors sample sizes downwards bias greater large positive values might expected alternative marginal likelihood estimation close discussion alternative approach binary time series regression modelling proposed cui modified glm mglm method replaces exp xtnt exp xtnt glm function xtnt representing marginal mean arrive objective function exnt log xtnt log xtnt xtnt exnt mglm estimater found iterating two steps starting glm estimate step estimate curve step maximize respect based estimate obtained first step steps repeated iteration stops maximum value reached last update regarded mglm estimator implementation details provided cui defining method cui require distribution latent process hence required strictly increasing however main theorem concerning consistency asympototic normality stated terms estimation binomial time series mixed models latent process specification specifications application method estimating requires additional constraintsrwhich currently implemented example taking first derivative respect gives application jensen inequality applied boat race time series estimate monotonic produces marginal estimates values gaps observed values xtnt zero therefore useful prediction new values linear predictor although implemented cui constraint well monotonicity enforced nonparametric estimation using alternative local linearization used cui example constraints monotonicity different estimates compared cui observed model boat race series analyse example marginal likelihood estimates give hence degenerates glm estimate differs cui somehow marginal method without monotonocity contraints avoiding degeneracy issue arises marginal estimation method proposed paper needs understood mglm computationally much intensive using standard glmm methods obtaining marginal estimates appears avoid degeneracy reasons fully understood stage additionally clear extent mglm without proper constraints implied latent process specification avoids high proportion degenerate estimates observed glmm binary data additionally extent mglm reproduces correct curve marginal probabilities true data generating mechanism defined terms latent process parameter driven specification investigated extent mglm estimate differs curve defined latent process specification might form basis test distribution latent process gaussian true curve evaluated glmm fits may end different results estimates discussion overcome inconsistency glm estimates regression parameters parameter driven binomial models time series models proposed use marginal likelihood estimation easily conducted using generalized linear mixing model fitting packages shown estimates regression parameters latent process variation obtained method consistent asymptotically normal even observations serially dependent distribution marginal estimates required score test serial dependence latent process something report elsewhere asymptotic results proofs thereof assumed latent process gaussian helped streamline presentation required results except lemma asymptotic identifiability binary case relies directly normal distribution proofs readily modified provided assume moment generating function finite defines parameter space structure model considered theoretical results apply response distributions poisson negative binomial little change proofs theorems glm estimation cases consistent asymptotically normal regardless serial dependence latent process true use marginal estimation advantage latent process variability also estimated yet shown expect response distributions use marginal estimation lead efficient estimates regression parameters response distributions moderate sample sizes marginal estimation method result probability observed simulations explained via theoretical asymptotic arguments particularly problematical binary responses high probabilities observed expected theory probability observed binomial data probability quickly decreases zero result observation anticipate probability typically large poisson negative binomial responses estimation binomial time series mixed models binary data developed useful upper bound approximation probability subsequently proposed improved mixture distribution finite samples theoretical derivations well supported simulations presented mixture distribution used based single time series provided useful insights sampling properties marginal estimation binary time series additionally derivations suggest regression models varies interval inverse logit function approximately linear particularly prone problem persists even strong serial dependence practitioners apply marginal likelihood method caution situations acknowledgements thank cui providing application mglm method boat race data reported cui appendix proof lemma consider deterministic regressors proof arguments extended stochastic regressors largest value log log since integrand integrand zero integrand zero almost everywhere hence contradiction assume log log happen since straightforward show iterating equivalent next show way hold fix denote expectation respect density hence equivalent assume exists holds estimation binomial time series mixed models zero time equivalent matrix full rank full rank ratios second column first column however show ratio increases increases since functions define probability densities egj egj egj denotes expectation respect since increasing function positively correlated therefore egj egj follows holds unique solution contradicts assumption thus holds implies condition conclude implies therefore condition holds proof lemma proof considers fourier transform method used wang louis assume connected first derivative respect sides also implies exists constant rewritten convolutions exp let fourier transform normal density fourier transform logistic density using fact fourier transform convolution product fourier transform function duh estimation binomial time series mixed models follows fourier transform mean zero normal distribution exp fourier transform logistic distribution thus exp exp sinh sinh fixed expressed function function constant contradicts definition hence equality holds appendix proof theorem proof presented three steps first show defined converges defined condition respectively second defined using compactness parameter space follows third proof use jensen inequality multiple times xtt max xtt pmt conditional bounded condition regressor xnt nonrandom marginal density strong law large numbers mixing processes mcleish applied gives lim log lim log defined condition ergodic properties stationary processes used establish log lim log lim defined consistency write log blais macgibbon roy proposition strongly mixing apply strong law large numbers mixing process mcleish need show kqt eqt denotes norm minkowski inequality inequality kqt eqt kqt estimation binomial time series mixed models using similar derivations bounded used suffices establish condition xnt bounded given therefore result follows condition given eqt together first part proof given since compact set continuous function gallant white theorem arg max asymptotic normality using taylor expansion asymptotic normality obtained conditional xnt strongly mixing chebyshev inequality ibragimov theorem since continuity respect hence follows next show positive definite let dimensional constant vector without lose generality define note det det xnt condition positive definite limit condition given stationary regressors limit easily obtained using ergodic theorem next show exists note lim var cov cov exists lim lim since strong mixing theorem ibragimov estimation binomial time series mixed models lim kqt exists example take use inequality xnt xnt xnt xnt application clt mixing process theorem davidson theorem following proof davis dunsmuir wang cui let note centre true value done references maximizing xtnt xtnt equivalent minimizing let arg min lim write xtt xtt xtt xtt xtt xtt using similar procedures proof theorem cui straightforward show exp since convex function minimizes application functional limit theory gives arg min lim conclusion space proof theorem since linear function show normally distributed sufficient show joint distribution multivariate normal defined xnt cnt cnt xnt defines sequence vectors let unt joint vector time dimension unt uniformly bounded strongly mixing unt need prove normal distribution arbitrary estimation binomial time series mixed models constant vector without loss generality slln mixing process mcleish exists limiting matrix var unt cov unt cov unt conditions clt davidson satisfied unt proof theorem proof follows moran true value parameters limit parameters maximize fixed maximum likelihood estimators consider first distribution since unconditional joint distribution multivariate normal see proof theorem hence distribution skew normal based use taylor expansion first derivatives around conditional references blais macgibbon roy limit theorems regression models time series counts statistics probability letters cox statistical analysis time series recent developments scandinavian journal statistics davidson central limit theorem globally nonstationary dependent functions mixing processes econometric theory davis dunsmuir wang autocorrelation poisson regression model biometrika davis dunsmuir state space models count time series crc monographs davis negative binomial model time series counts biometrika dunsmuir tran weatherburn assessing impact mandatory dna testing prison inmates nsw clearance charge conviction rates selected crime categories nsw bureau crime statistics research gallant white unified theory estimation inference nonlinear dynamic models basil blackwell new york gupta multivariate skew normal distribution journal multivariate analysis heagerty lumley window subsampling estimating functions application regression models journal american statistical association ibragimov independent stationary sequences random variables woltersnoordhoff mcleish maximal inequality dependent strong laws annals probability estimation binomial time series mixed models moran maximum likelihood estimation conditions mathematical proceedings cambridge philosophical society neuhaus kalbfleisch hauck comparison approaches analyzing correlated binary data international statistical internationale statistique wang louis matching conditional marginal shapes binary random intercept models using bridge distribution function biometrika variance estimation negative binomial time series regression model journal multivariate analysis cui logit regression model binary time series journal time series analysis
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jsc javascript object system artur ventura dec january abstract jsc language superset javascript designed ease development large web applications language extends javascript object system isolating code class declaration simplifying multiple inheritance using method implementation agreements motivation web applications gaining dynamic behavior javascript become important web development useful use software engineering aproach javascript javascript model model extremely versatile develop however large team developers problems name colisions unintentional method redefinition occur also maintenance becomes issue jsc attempts remedy issues implementing object system top standard javascript related work attemps add structed programming javascript ecmascript language witch javascript dialect last version added support defining classes another dialect ecmascript actionscript supports defining classes althought systems advanced support typechecking via anotation variables generics dynamic classes templates etc avaible current browsers expected future versions javascript extends behavior ecmascript attempt creating creating supersets javascript one example sctrict superspet javascript adds object system implementing language similar aproach similar jsc language uses message passing making methods unparsable javascript object oriented javascript javascript class function definition similar definition function shown next function rectangle function javascript type function behaves similarly hash dictionary one properties object prototype property represents structure yet fully defined formed instantiated instantiation done new rectangle new operator invoke function rectangle new object reference return changes made object afect new object prototype property defined object prototype prototype value returned method defined adding function class prototype shown next function return methods called like constructor getheight invoked reference instance case rectangle method avaible instances define localy method getarea code shows inheritance implemented copying prototype superclass subclass function positionedrectangle new rectangle actually rectangle could return anything default problems first problem inexistence seperation functional object oriented programming examples shown extensions javascript functional programming allows create behavior similar object oriented another problem today browsers share enviroment javascript files possible method defined two distinct files difficult debugging maintenance objectives jsc extension javascript developed following goals mind create new language keep much possible javascript syntax semantics provide mechanisms maintain code isolated packages provide protocol allows easy use reflection intersection keeping code organized provide simple easy use multiple inheritance provide implementation agreement classes similar java interfaces protocols jsc class definition jsc defined classes class defined single file simple jsc class show next package class rectangle slots height width rectangle function getarea function return class begins package declaration class always referred package class name case body class declared javascript object fourth line represents slot declaration jsc class slots directly used access slot getter setter method provided slot fifth ninth lines declare constructor method respectively constructor method name class constructor required one class important notice class header slot definition capable parsed javascript instantiation jsc attempts minimize usage global enviroment actually global definition required function class function expects string canonical class location package name returns representing class example rectangle obtained class runtime contains among others method called create method create new instance call class constructor instantiate rectangle class class inheritance inheritance used extending superclass shown next package class positionedrectangle extends slots positionedrectangle function class inheritance jsc works like mixins class method slot superclass added subclass class extend several classes positionedrectangle height slot getters setters rectangle copied token extends insted mixin used makes class header similar ecmascript java object contains init method calls constructor class using first argument instance similar python superclass constructor called static enviroment jsc class declare methods used level simple example shown next package main class app static main function args protocols protocol jsc assures existence certain methods protocol declaration shown next package protocol draggable element true eventlistener false example draggable declares existence two methods element eventlistener keyword next method name declares need implement true declares method required implemented class extends protocol false guarantees method implemented empty function name provided protocol extended protocols class implement number protocols verification required methods done class initialization jsc classes classpool going initialized done invoking method classinit method compute effective set methods slots class possesses detect protocols satisfied finally setup prototype see usage details runtime intersection possible create new change altering definition moment execution however required call classinit prototype reconstructed also possible extend behavior jsc extending class protocols one example quite simple implement something similar java abstract classes jsc current problems global variables jsc attempts minimize usage global enviroment encapsulating code classes however however javascript allows declaration gobal variables within function variable declared without keyword var declared global enviroment instance function foo var local global currently allowed jsc verified slot default value currently slot declaration allow slot default value would nice slot declaration similar package bar class foo slots aslot getter getslot setter setit anotherslot default class currently jsc support syntax easy use parser javascript available jsc compiler assures correctness code loading methods javascript engine detecting parsing errors parser available global variables problem addressed jscc jsc compiler currently uses google engine usage current compiler target usage methods server jsc developed due need developing lot code run web server mode code loaded rdbms loaded upon need client another usage generating single classpool image self contained javascript file load browser like ordinary javascript file jsc code accessed javascript files using class function jsc virtual machine jscvm small virtual machine implementation jsc performance small library implemented jsc files occupies compilation targeting browser ended single file occupying file takes second chrome seconds firefox using yui file shrinked almost loaded instantaneously chrome firefox yui compressor available http application removes whitespaces also reduces variable names
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mean variance phylogenetic trees david daniel megan aug cheriton school computer science university waterloo waterloo canada department mathematics lehman college city university new york new york usa corresponding author megan owen department mathematics lehman college city university new york bedford park blvd west bronx usa describe use mean variance bhv treespace summarize explore diversity set phylogenetic trees show mean comparable summary methods despite stickiness property variance faster precise commonly used variance measures mean variance measures theoretically justified robust previous estimates type estimated reasonably efficiently leads applications mean hypothesis testing keywords phylogenetics treespace mean variance test sets related phylogenetic trees commonly encountered evolutionary biology example one might encounter set output inference program like mrbayes ronquist set gene trees given set species paper think set sample underlying distribution set phylogenetic trees fixed leaf set consider basis describing mean distribution using representation trees elements geometric space looking mean tree minimizes sum squared distance mean elements sample distribution formulation also allows identification variance actual sum squared distances sample elements mean describe algorithms computing mean variance fairly efficient compute show mean gives good way summarizing uncertainty found set trees particularly nice property mean may result tree statistically appropriate way also describe computation variance variance better measure statistical uncertainty simpler measures like number topologies found set trees faster compute sum pairwise distances trees large set trees results show measure phylogenetic distance originated billera fact used practical applications background begin describing treespace working properties space distance measure also consider ways summarizing collection trees besides mean variance properties treespace bhv treespace billera contains unrooted phylogenetic trees edge lengths given set labelled leaves paper fix set leaf labels trees thought rooted fixing leaf root paper defined bhv treespace subspace number possible splits leaves equivalently number possible partitions set leaves two sets size least coordinate corresponds different split order splits matter fixed note allowing partitions size include splits corresponding edges ending leaves original definition billera ignores edges notes included done given tree leaves edge lengths corresponds following vector every edge length let coordinate corresponding split induced let coordinates corresponding splits induced edges let set vectors correspond trees described vectors correspond trees due split incompatibility two splits incompatible induced edges existing tree example cherry pair adjacent leaves tree corresponding split separates two adjacent leaves others tree cherries corresponding splits incompatible vectors positive values coordinates topology tree set splits induced edges tree binary tree interior nodes degree contains splits degenerate trees contain fewer splits see figure details including combinatorial description space see billera distance metric associated bhv treespace called bhv distance geodesic distance also defined billera consider trees topology trees correspond set splits vectors exactly set coordinates take positive values thus number coordinates set trees corresponds euclidean orthant part figure five three fifteen orthants bhv treespace ease visualization five dimensions corresponding leaf edges included thus binary tree topology represented quadrant orthant two quadrant axes corresponding lengths interior edge splits axis two quadrants corresponds tree topology fully resolved geodesic shortest path two trees shown dashed line may pass different orthants depending branch lengths endpoint trees shown geodesics tree topology binary two trees topology bhv distance euclidean distance corresponding vectors orthant two trees different topologies bhv distance length shortest path remains treespace length path computed calculating euclidean distance path restricted orthant passes though summing lengths shortest path called geodesic pass one orthant next orthant boundaries corresponding trees fewer splits see figure example two geodesics bhv treespace connected since two trees connected path origin may may geodesic billera showed bhv treespace globally curved bridson haefliger implies geodesics unique owen provan gave polynomial time algorithm computing geodesic distance two trees runs time faster trees common splits note metric also known manhattan taxicab metric used orthant instead euclidean metric length longer unique geodesic two trees weighted distance robinson foulds see john good summary mean variance euclidean space mean centre mass point minimizing sum squared distances sample points equivalent average sample points mean similarly defined treespace miller independently set sample trees mean simply mean tree minimizes bhv distance two trees variance simply variance minimum sum squared distances mean unique treespace curved miller gave algorithm approximating mean variance based law large numbers curved space derived sturm briefly mention interesting properties mean see miller section details proofs mean tree necessarily refinement consensus tree contains splits found majority trees split appears mean tree appears least one sample trees split appears sample trees appears mean tree finally mean tree sticky hotz perturbing one sample trees always change mean tree stickiness happens mean orthant treespace corresponds degenerate tree topology implies mean unresolved often one might expect wish similar consensus tree coincidence consensus tree median tree distance mcmorris pattengale showed weighted consensus tree median weighted distance recall using metric orthant space instead euclidean metric gives weighted distance instead bhv distance variance set trees quantifies spread set trees mean show experiments sequence length increases information tree reconstructed variance samples trees bootstrap posterior distributions decreases sturm showed law large numbers holds meaning sample size increases sample means distribution converge true mean barden proved central limit theorem bhv treespace showing distribution sample means converges certain gaussian distribution measures centre among set trees compare mean three commonly used measures centre consensus phylogenetics first tree tree containing exactly splits appearing majority input trees margush mcmorris consensus tree mean general refinements miller two measures centre comes tree search algorithms cases tree search procedures produce distribution trees along probable tree acts like centre specifically look maximum likelihood search using raxml stamatakis produces distribution bootstrap trees felsenstein maximum likelihood tree considered centre alternatively using bayesian approach phylogenetic inference implementation mrbayes ronquist returns posterior distribution representing probable tree maximum posteriori tree map tree often used summary tree measures variance variance amount variability set trees less established measure summary tree often visual representation splitstree huson bryant densitree bouckaert used represent diversity set trees unfortunately methods assessed compared quantitatively instead compare variance several quantitative measures variance namely number different tree topologies set number different splits set sum squared geodesic distances pair trees set later measure proposed chakerian holmes method approximating variance used ponciano measure variance posterior distributions requires time estimate variance trees leaves one also explore variability set trees using pairwise comparison methods applied pairs trees set one compare trees using bhv distance distance robinson foulds distance trees defined number splits plus number splits size symmetric difference split sets note definition uses tree topologies edge lengths trees related work closely related work benner investigated behaviour mean median tree minimizing sum distances instead sum squared distances sample trees summarize posterior distributions returned bayesian tree reconstruction methods authors simulated sequences lengths using tree plants evolutionary model felsenstein show mean median estimates comparable consensus estimate instances perform better also investigate variance changes sequence length work conceived independently overall aim work scope experiments much broader considering tree distributions generated maximum likelihood bayesian methods general gtr evolutionary model trees taxa comprehensive look variance paper focuses mean variance set trees point summary data work look summaries best fit lines bhv treespace nye feragen extensions summaries generalization principal components analysis pca nye confidence sets bhv treespace constructed willis full central limit theorem given barden methods computation mean variance mean tree tree minimizes sum squared bhv distances tree sample trees equivalent centre mass euclidean space compute approximation mean tree using iterative implementation described miller implementation new approximation mean tree returned iteration approximations converge true mean tree number iterations grows decide stop iterative algorithm use program option check convergence using cauchy sequence length epsilon means stop iterative algorithm pairwise bhv distances last mean approximations less equal experiments means converged within iterations also use random permutation heuristic validated choice epsilon choosing one repetition sequence length computing approximate mean corresponding posterior distribution sample times using chosen parameters epsilon average bhv distance pairs approximate means sample order considered acceptable computing mean variance took minutes ghz intel xeon processor program source code available http simulated data opossum diprotodontian sloth anteater armadillo hedgehog shrew mole phyllostomid freetailedbat falsevampirebat flyingfox rousettefruitbat whale dolphin hippo ruminant pig llama horse rhino tapir cat caniform pangolin sciurid mouse rat hystricid caviomorph rabbit pika flyinglemur treeshrew strepsirrhine human tenrecid goldenmole shortearedelephantshrew longearedelephantshrew aardvark sirenian hyrax elephant figure reference mammal tree murphy simulated sets dna sequences variety lengths base pairs using tree mammals murphy shown figure reference tree sequences simulated version rambaut grass using gtr model following parameter settings proportion invariable sites equilibrium frequencies respectively gtr rate matrix entries transition rate parameter values estimated hillis reference tree set sequences length ran raxml version stamatakis compute maximum likelihood tree bootstrap sample trees raxml settings used gtr evolutionary model conducted full analysis option using rapid bootstrapping option recommended raxml user manual computing large number bootstrap replicates since every bootstrap tree used starting point tree search set sequences length also ran mrbayes ronquist compute map tree sample trees posterior distribution mrbayes settings used evolutionary model ran iterations sampling every generations used last sampled trees posterior distribution sample distribution trees estimated mean variance using sturmmean miller settings explained previous subsection also computed consensus tree three measures variation sample number different tree topologies sample number different splits sample sum squared bhv distance pair trees sample compared mean tree consensus map trees follows sequence length repetition distribution type bootstrap posterior one mean tree one consensus tree either map tree depending whether sample bootstrap posterior distribution adding reference tree set computed distance pair trees also computed bhv distance trees pair meaningful branch lengths mean trees trees reference tree meaningful branch lengths consensus tree map trees also compared lengths sampled centres define trees length tree edge set bhv distance tree bhv treespace origin investigate stickiness mean compare length mean tree distribution average length trees distribution see mean much shorter component trees also compare lengths mean average lengths trees posterior bootstrap distributions generated sequence set cases use wilcoxon ranked signed test wilcoxon test hypothesis two distribution lengths finally conducted mean hypothesis test bootstrap posterior samples first repetition base pairs recall mean hypothesis test type test tests means two samples rejecting hypothesis implies samples different distributions computed bhv distance means bootstrap posterior samples compared bhv distance means random partition two samples since full permutation test feasible since sample contains trees estimated using randomly chosen permutations note following method performing hypothesis testing trees suggested feragen results visualize trees one repetition experiment look like first repetition base pair sequence length experiment computed bhv distance trees bootstrap posterior samples reference tree tree two mean trees use classic scaling mds kruskal reduce two dimensions fig two means tree middle respective samples two samples separated space reference tree closer posterior sample near centre reference tree two means tree trees two samples topology common topologies appear bootstrap posterior samples suggests difference two clusters figure primarily due branch lengths bhv distance takes account instead topology reference tree bootstrap tree posterior tree bootstrap mean posterior mean figure sample trees first repetition base pair sequence length embedded dimensions using classic scaling trees bootstrap posterior samples form two clusters likely due differences branch lengths instead topology comparison mean tree measures centre first compare mean trees reference tree summary trees showing close four plots figure use bhv distances compare mean bootstrap posterior sample corresponding consensus tree reference tree either map tree appropriate cases distance mean trees decreases sequence length increases mean tree closer map consensus tree reference tree three trees almost always topology sequences length base pairs longer next consider relation reference tree reconstructed trees four plots figure use bhv distance compare reference tree mean tree consensus tree map tree sample plots combined previous figure show reconstructed trees closer reference tree expected trees become closer reference tree sequence length increases since gives information reference tree improving reconstruction interestingly bhv distance mean tree bootstrap distribution average closer reference tree tree variance compare different measures variance set trees namely number different topologies number different splits variance sum squared bhv distance pairs input trees figure sequence length increases information underlying tree expect raxml mrbayes better job inferring tree certain reflected decrease variance bootstrap posterior distributions thus samples sequence length increases see figure measures posterior samples lower variance bootstrap samples matches previous observations posterior probabilities higher bootstrap frequencies clades erixon douady huelsenbeck rannala since trees lower variance sample spread thus fewer different splits trees higher variance sample general find uncertainty variance estimate metrics variance measurements sum squared bhv distance pairs sample trees variance almost identical expected two measures identical euclidean space sum squared bhv distance pairs first suggested measure variance large input sets faster compute variance bhv distance pairs see discussion details bootstrap distribution posterior distribution reference consensus reference map consensus distance mean tree distance mean tree sequence length sequence length bootstrap distribution posterior distribution refence reference bhv distance mean tree bhv distance mean tree sequence length sequence length figure sample bhv distances calculated mean tree tree map tree reference tree distribution distances shown using box plots mean tree approaches map consensus trees sequence length increases three trees usually sharing topology sequence lengths base pairs bootstrap distribution posterior distribution mean consensus mean map consensus distance reference tree distance reference tree sequence length sequence length bootstrap distribution mean bhv distance reference tree sequence length figure sample bhv distances calculated reference tree sample mean tree tree map tree distribution distances shown using box plots three reconstructed trees approach reference tree sequence length increases mean tree slightly closer average tree bhv distance measure bhv distance reference mean trees posterior distribution shown graph figure bootstrap samples posterior samples bootstrap samples posterior samples number different splits number different topologies sequence length sequence length bootstrap samples posterior samples bootstrap samples posterior samples frechet variance sum squared bhv distance pairs sequence length sequence length figure variance samples bootstrap posterior distributions measured using number different topologies number different splits sum squared bhv distance pairs variance measures posterior samples lower variance average bootstrap samples sum squared bhv distance pairs variance similar note axis number different splits sum squares variance log scale tree length comparison also looked average tree lengths determine effect mean stickiness mean trees shorter sequence lengths fully resolved due stickiness phenomenon mean trees base pair sequence length fully resolved binary however even mean trees fully resolved stickiness property could still cause closer origin sample expected terms measurable effects would translate mean trees shorter terms total edge length average trees computed matches found bootstrap posterior samples length mean tree strictly less average length corresponding sample trees cases less mean trees shorter length might expected noticeably affect quality summary average bootstrap posterior mean trees respectively average lengths respective samples also compared lengths trees bootstrap samples corresponding mean trees datasets length tree greater length corresponding mean tree length tree greater average length trees corresponding bootstrap sample interestingly average length trees posterior sample always less average length trees corresponding bootstrap sample generated set sequences suggests bootstrap samples spread could also reflect posterior probabilities higher clades bootstrap probabilities unsurprisingly mean tree posterior distribution sample always length less mean tree corresponding bootstrap sample mean hypothesis test first repetition base pair sequences performed hypothesis test compare samples bootstrap posterior distributions using means null hypothesis stated two means bootstrap posterior distributions tested hypothesis using approximate permutation test cases distance means random partition two samples strictly less distance means two samples thus reject null hypothesis estimated confidence interval means assume two distributions mean thus expected due previous work showing bootstrap posterior probabilities different erixon douady huelsenbeck rannala discussion experiments demonstrate mean variance behave expected way biological data shown mean samples bootstrap posterior distributions comparable accuracy map trees respectively well consensus tree indeed results seem suggest original reference tree slightly closer mean summary trees possible gain may enough warrant cost computing mean results conclusively demonstrate mean valid summary method sample trees believe value mean comes sound mathematical backing enables sophisticated statistical tests like mean hypothesis testing results variance experiments clear variance faster precise measure variance existing alternatives variability number topologies tree sets sequence length high comparison variance measures although variability decreases measuring number splits tree set still higher sum squared pairwise distances variance comparable sequence lengths sum squared pairwise distances variance similar profiles justifying use sum squared pairwise distances measure variance literature however large sample sizes variance faster compute compute sum squared geodesic distances trees one must compute geodesic distances contrast computing variance involves calculating geodesic distance per iteration algorithm number iterations required depends desired precision mean however often obtained good results iterations suggesting computing variances approximately trees variance calculation faster calculation experiments tree lengths show mean trees shorter average length sample trees less average demonstrates stickiness leads shorter mean trees difference lengths likely significant ignored one might think could use mean compute species tree set gene trees miller example even restrict coalescent model explaining gene tree diversity still problematic two regards coalescent model pendant edges gene trees must least long pendant edges species tree usually longer since pendant edge mean tree computed averaging length edge input trees pendant edges mean gene trees least long usually longer pendant edges true species tree possible topology mean tree might still match species tree conjectured miller example preliminary experiments suggest stickiness greatly limits amount information mean tree compared methods preliminary experiments mean tree set simulated gene trees became essentially degenerate gene trees still far away anomalous zone degnan rosenberg common gene tree topology still species tree topology essentially degenerate mean approximated mean tree within small distance degenerate tree possible exact algorithm computing mean would give mean tree topology even cases iterative algorithm certain mean degenerate even would informative conclusion shown mean variance behave expected way biological data shown mean used mean hypothesis testing compare two samples trees variance stable reliable measure amount variability sample trees validation opens door new applications quantities one possible application variance determine mrbayes markov chain monte carlo mcmc algorithms converge preliminary experiments mrbayes runs data paper show comparing variance sliding windows sampled trees identify variance trees sliding window remains roughly period however may datasets variance continue decrease indicating lack convergence iterative algorithm computing mean easily adapted online algorithm iteration instead choosing tree random input set trees use next tree generated mcmc chain unfortunately clear take advantage computing variance requires computing bhv distance current mean approximation sample trees seen far however algorithms computing geodesics dynamically skwerer provan might help references computing medians means hadamard spaces siam journal optimization barden logarithm map limits frechet means orthant spaces arxiv preprint mcmorris median procedure journal classification benner bourguignon point estimates phylogenetic reconstructions bioinformatics billera holmes vogtmann geometry space phylogenetic trees advances applied mathematics bouckaert densitree making sense sets phylogenetic trees bioinformatics bridson haefliger metric spaces curvature chakerian holmes computational tools evaluating phylogenetic hierarchical clustering trees journal computational graphical statistics degnan rosenberg discordance species trees likely gene trees plos genet douady delsuc boucher doolittle douzery comparison bayesian maximum likelihood bootstrap measures phylogenetic reliability molecular biology evolution erixon svennblad britton oxelman reliability bayesian posterior probabilities bootstrap frequencies phylogenetics systematic biology felsenstein evolutionary trees dna sequences maximum likelihood approach journal molecular evolution felsenstein confidence limits phylogenies approach using bootstrap evolution pages feragen owen petersen wille thomsen dirksen bruijne statistics approximations analysis anatomical trees pages ipmi vol hillis heath john analysis visualization tree space systematic biology hotz huckemann marron mattingly miller nolen owen patrangenaru skwerer sticky central limit theorems open books annals applied probability huelsenbeck rannala frequentist properties bayesian posterior probabilities phylogenetic trees simple complex substitution models systematic biology huson bryant application phylogenetic networks evolutionary studies molecular biology evolution kruskal multidimensional scaling optimizing goodness fit nonmetric hypothesis psychometrika margush mcmorris consensus bulletin mathematical biology miller owen provan polyhedral computational geometry averaging metric phylogenetic trees advances applied mathematics murphy eizirik brien madsen scally douady teeling ryder stanhope jong resolution early placental mammal radiation using bayesian phylogenetics science nye principal components analysis space phylogenetic trees annals statistics pages nye algorithm constructing principal geodesics phylogenetic treespace transactions computational biology bioinformatics nye tang weyenberg yoshida principal component analysis locus mean space phylogenetic trees arxiv preprint owen provan fast algorithm computing geodesic distances tree space transactions computational biology bioinformatics tcbb pattengale tools phylogenetic postprocessing thesis university new mexico ponciano burleigh braun taper assessing parameter identifiability phylogenetic models using data cloning systematic biology page rambaut grass application monte carlo simulation dna sequence evolution along phylogenetic trees computer applications biosciences cabios robinson foulds comparison weighted labelled trees pages combinatorial mathematics springer robinson foulds comparison phylogenetic trees mathematical biosciences ronquist teslenko van der mark ayres darling larget liu suchard huelsenbeck mrbayes efficient bayesian phylogenetic inference model choice across large model space systematic biology skwerer provan dynamic geodesics treespace via parametric maximum flow arxiv preprint john review paper shape phylogenetic treespace systematic biology stamatakis raxml version tool phylogenetic analysis large phylogenies bioinformatics sturm probability measures metric spaces nonpositive heat kernels analysis manifolds graphs metric spaces lecture notes quarter program heat kernels random walks analysis manifolds graphs april emile borel centre henri institute paris france wilcoxon individual comparisons ranking methods biometrics bulletin willis confidence sets phylogenetic trees arxiv preprint
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may optimal prevention possibilistic mixed background risk irina georgescu academy economic studies department economic informatics cybernetics romana oficiul postal bucharest romania email ana lucia casademunt universidad department business administration cordoba spain email alucia abstract paper effect posibilistic mixed background risk level optimal prevention studied framework five purely possibilistic mixed models necessary sufficient conditions found level optimal saving decreases increases result actions various types background risk way results complete obtained courbage rey prevention models probabilistic background risk keywords possibilistic background risk optimal prevention optimal saving modal operator introduction prevention effort made agent reduce action unwanted event financial loss effort amount money invested agent prevention activities interested determine optimal level effort according prevention consists reducing probability loss size loss simplicity paper prevention one understands terminology used first contribution study optimal prevention paper elrich becker investigates interconnection market insurance subject generated real literature focused especially way consumers preferences influence optimal level prevention see common topic deals relationship risk aversion optimal prevention instance dionne eeckhoudt emphasize fact increase risk aversion lead increase optimal investment prevention effect ambiguous second topic relationship notion prudence introduced kimball optimal prevention see paper studies impact prudence optimal prevention model analyzes obtaining optimal prevention framework latter case prevention effort done first period effect likelihood appearance loss occurs second period prevention models background risk appear first time paper courbage ray eeckhoudt background risk may appear one periods respect changes optimal prevention level occur relation benchmark model background risk exist mentioned papers deal probabilistic models background risk represented random variable paper aims study optimal prevention models possibilistic mixed background risk see former case background risk fuzzy number latter case one period background risk random variable period fuzzy number section contains definition possibilistic expected utility properties used paper section benchmark prevention model presented following starting point background risk model construction next sections section eight background risk prevention models described constructions risk two periods may either random variable fuzzy number first three models probabilistic theory already developed five models purely possibilistic mixed construction follows similar line first three total utility functions defined first order conditions verified optimal prevention deduced section focuses model various types background risk five models influence level optimal prevention five models level optimal prevention compared benchmark model necessary sufficient conditions increase decrease optimal prevention respect existing level case benchmark model proved section background risk effect optimal saving studied already existing risk one period possibilistic probabilistic risk added period possibilistic expected utility section recall definition possibilistic expected utility basic properties random variable defined probability field denote expected value continuous utility function expected value random variable probabilistic expected utility associated develop possibilistic risk theory need similar notion possibilistic expected utility definition assumes framework composed following components utility function representing agent fuzzy number modelling possibilistic risk weighting function andrmonotone increasing function satisfying normality condition assume level sets fuzzy number form possibilistic expected utility defined identity function exactly possibilistic expected value following two propositions emphasize two important properties possibilistic expected utility first one linearity property second one jensen type inequality used following analysis optimal prevention variations respect various types background risk proposition let two continuous utility functions aef proposition utility function convex benchmark prevention model section present according entities compose benchmark prevention model form total utility function condition level optimal prevention selected benchmark prevention model two periods prevention effort conducted period effect effort probability occurrence loss wealth manifested period model defined following entities safe wealth periods utility functions period respectively level prevention effort period loss level probability loss occurs prevention level model studied hypotheses utility functions class level according initial data second period consumption activity endogenous risk parameterized prevention level whose outcomes expected utility corresponding second period total utility function agents wish choose level effort maximizing total utility comes finding solution optimization problem max since strictly concave optimal solution given leads condition prevention models background risk prevention models section built starting benchmark model section definition founded following assumptions assumption background risk may appear period period assumption background risk period modeled probabilistically random variable possibilistically fuzzy number denoting lack background risk one two periods eight models defined basis two assumptions specified table period probabilistic background risk probabilistic background risk possibilistic background risk possibilistic background risk probabilistic background risk possibilistic background risk period probabilistic background risk probabilistic background risk possibilistic background risk possibilistic background risk possibilistic background risk probabilistic background risk cases deal probabilistic models cases possibilistic models last two cases mixed models models studied next present briefly construction models model period appears probabilistic risk modeled random variable total utility function following form first derivative optimal value given condition model period appears probabilistic risk modeled random variable overall utility function first derivative optimal value given condition model background risk appears periods modeled random variables respectively total utility function form first derivative optimal value given condition next present construction models table rest section fix weighting function model assume background risk described fuzzy number whose level sets total utility function model according definition possibilistic expected utility written derivation one obtains following form derivative follows deriving one time one reaches hypotheses imposed utility functions probability follows level prevention effort thus concave function value maximizing total utility function verifies condition written model assume background risk period fuzzy number whose level sets total utility function model derivation one obtains due monotonicity property possibilistic expected utility inequality therefore hence strictly concave value maximizing verifies condition written model assume background risk period fuzzy number background risk period fuzzy number total utility function model derivation one obtains last inequality follows thus strictly concave value maximizing verifies condition written model assume background risk period random variable background risk period fuzzy number total utility function model derivation one obtains follows level effort level thus value maximizing given condition written model assume background risk period fuzzy number background risk period random variable total utility function model derivation one obtains one proves immediately concave thus value maximizing verifies written remark models incorporate entities benchmark model features background risk types defining appear expression total utility functions condition determines level optimal prevention effect background risk optimal prevention section study way adding various types background risk may produce changes level optimal saving compare optimal prevention level benchmark model optimal prevention levels eight models previous section models already treated handle models table case situated framework set defined section case comparison following result establishes necessary sufficient condition adding possibilistic background risk period optimal saving level drops proposition following assertions equivalent proof previous section know thus strictly decreasing function following equivalences hold iff iff iff iff corollary proof applying proposition convex function follows hence proposition one obtains corollary shows agent prudent period presence possibilistic background risk first period reduces optimal prevention notion prudence possibilistic mixed framework studied case comparison following result establishes necessary sufficient condition adding possibilistic background risk period optimal prevention level raises proposition following assertions equivalent proof function strictly decreasing thus applying following equivalences hold iff iff iff iff lemma proof consider function assumption follows strictly increasing therefore follows function convex thus applying proposition noticing one obtains desired inequality corollary proof proposition lemma applied case comparison following result establishes necessary sufficient condition adding possibilistic background risk period possibilistic background risk period optimal prevention level raises proposition following assertions equivalent proof function strictly decreasing thus applying following assertions equivalent taking account follows immediately equivalence assertions remark assume reasoning analogous one corollary follows applying lemma follows two inequalities proposition conclude prudent agent periods effect possibilistic background risks optimal prevention ambiguous establish positioning levels one case comparison following result establishes necessary sufficient condition adding probabilistic risk period possibilistic risk period optimal prevention level raises proposition following assertions equivalent proof analogous proof proposition using fact strictly decreasing applying remark assume follows according jensen inequality following inequality lemma conditions imply inequality two inequalities show background risk effect described mixed vector optimal prevention ambiguous case comparison following proposition establishes necessary sufficient condition adding possibilistic background risk period probabilistic risk period optimal prevention level raises proposition following assertions equivalent proof applied remark assume proposition following inequalities hold thus effect mixed background risk level optimal saving ambiguous cases comparing optimal prevention levels previous section studied changes optimal prevention pass benchmark model models background risk appears one periods periods problem changes optimal saving posed cases one passes situation background risk present single period models one situations background risk present periods models purpose compare optimal prevention levels following pairs section deal problem making comparison optimal prevention levels mentioned pairs remaining cases treated similar manner case comparison one assumes period exists background risk described fuzzy number added period possibilistic background risk described fuzzy number following proposition offers necessary sufficient condition change optimal prevention level drops proposition following assertions equivalent proof taking account following assertions equivalent corollary proof lemma condition proposition fulfilled previous corollary shows prudent agent period adding possibilistic background risk already existing possibilistic background risk leads raise optimal prevention case comparison one considers model possibilistic background risk period intend see optimal prevention changes pass model adding possibilistic background risk period proposition following assertions equivalent proof following equivalences hold iff iff iff iff corollary proof applying proposition follows condition iii proposition verified case comparison one considers model probabilistic background risk period one passes model adding possibilistic background risk period proposition following assertions equivalent proof recall section model following assertions equivalent corollary proof condition proposition fulfilled concluding remarks five models paper deal optimal prevention presence risk situations possibilistically described fuzzy numbers mixed combinations random variables fuzzy numbers come addition probabilistic models optimal saving developed courbage rey main results paper establish necessary sufficient conditions optimal prevention increase decrease add possibilistic mixed risk benchmark model next state several research topics could investigated line paper among entities define benchmark model section probability essential role also eight background risk models section contain probability defining element accordingly even background risk possibilistic mixed presence model still preserves probabilistic trace could modify nature following way case possibility loss occurs prevention level case credibility loss occurs prevention level study optimal prevention models possibility case credibility case respectively definition prevention models total utility function order condition etc optimal prevention level comparison theorems formulated terms possibilistic expected utility probabilistic expected utility hand literature exists second notion possibilistic expected utility see example section well concept expected utility operator see chapter comprises generalize prevention models paper general context founded expected utility operators definition credibility properties see monograph chapter iii order reduce action various types risk agent within reach several instruments saving according precautionary motive agent defined extent agent chooses instrument response future risk paper analyzed way intensity precautionary motives influenced choice various combinations instruments well interaction considered types risk open problem develop models precautionary saving analysis corresponding combinations three instruments saving risk parameters probabilistic possibilistic mixed references carlsson possibilistic mean value variance fuzzy numbers fuzzy sets syst carlsson possibility decision springer georgescu optimal saving prudence possibilistic framework dcai volume advances intelligent systems computing courbage precautionary saving presence risks economic theory courbage rey treich prevention precaution dionne handbook insurance huebner international series risk insurance economic security new york springer new york courbage peter optimal prevention multiple risks risk insurance press dionne eeckhoudt increased risk aversion economics letters dubois prade fuzzy sets systems theory applications new york academic press dubois prade possibility theory new york plenum press eeckhoudt gollier schlesinger economic financial decisions risk princeton university press eeckhoudt gollier impact prudence optimal prevention economic theory eeckhoudt huang tzeng precautionary effort new look risk uncertainty ehrlich becker market insurance selfprotection journal political economy majlender weighted possibilistic mean variance fuzzy numbers fuzzy sets systems june georgescu possibility theory risk springer georgescu risk aversion prudence mixed optimal saving models kybernetika heinzel peter precautionary motives multiple instruments available http kimball precautionary saving small large econometrica liu uncertainty theory berlin
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jun bernoulli doi integral approximation kernel smoothing bernard institut recherches rennes irmar campus beaulieu rennes rennes france institut statistique biostatistique sciences actuarielles isba catholique louvain belgique let sequence random variables show function regularity conditions classical kernel estimator density result striking speeds traditional rates root derived central limit theorem although paper highlights applications mainly address theoretical issues related later result derive upper bounds rate convergence probability bounds depend regularity functions dimension bandwidth kernel estimator moreover shown accurate since used renormalizing sequences two central limit theorems reflecting different degrees smoothness application regression modelling random design provide asymptotic normality estimation linear functionals regression function consequence result asymptotic variance depend regression function finally debate choice bandwidth integral approximation highlight good behavior procedure simulations keywords central limit theorem integral approximation kernel smoothing nonparametric regression introduction let sequence random variables show function regularity conditions electronic reprint original article published bernoulli vol reprint differs original pagination typographic detail delyon portier classical kernel estimator density say defined every kernel called bandwidth needs chosen certainly depend result central limit theorem lead following reasoning estimating integral function evaluated random grid whether known using kernel estimator provides better convergence rates using result certainly consequences field integral approximation area many deterministic well random methods available accuracy respect computational time usual allows compare advantages random deterministic framework lie stability highdimensional settings comprehensive comparison approaches refer among random methods importance sampling widely used technique basically reduces variance classical integration good choice sampling distribution called sampler estimators unbiased form regarding mean squared error mse optimal sampler unique depends see theorem page among others parametric nonparametric studies focused estimation optimal sampler equation indicates new weighting observations weight reflects isolated point among sample therefore estimator takes account information giving weight isolated point summary procedure adaptive design points enjoys following advantages faster root rates estimation based unique sample drawn possibly unknown best knowledge design controlled rates obtained many semiparametric problems important issue construct root estimators possibly efficient rely kernel estimator nuisance parameter among others addressed stone case estimation location parameter robinson partially linear regression model stoker studying single index model result equation would seen superefficient estimator therle cam theory linked actually theory since quantity interest depend distribution result link work semiparametric literature relies mainly strategy employed substituting density kernel estimator paper propose comprehensive study convergence stated equation similar result originally stated vial chapter equation integral approximation kernel smoothing lemma context multiple index model best knowledge type asymptotic result addressed yet particular problem theoretical aim extend result precise upper bounds dimension window regularity impact bounds showing central limit theorems specifying regularity achieve program need introduce corrected version estimate bias reduced first corrected estimator shown better rates convergence initial one second shown asymptotically normal rates case regular rates special case jumps boundary support compute asymptotic distribution rely paper hall central limit theorem completely degenerate obtained important point succeeded proving result much weaker assumptions regularity regularity instance equation may hold even jumps however estimation subject curse dimensionality required smooth enough regarding dimension aim also link equation nonparametric regression random design model unknown particular obtain asymptotic normality estimators linear functionals thanks fast rates detailed previously asymptotic distribution depend function paper organized follows section deals technical issues related equation particular examine rates convergence according choice bandwidth dimension regularity functions section dedicated convergence distribution estimators section show apply equation problem estimation linear regression functionals finally section give simulations compare method traditional procedure integration proofs technicalities postponed section end paper rates convergences faster root section first provide upper bounds rates convergence probability estimators main purpose show rates faster root hold wide range parameter settings estimation second argue faster thanr root rates reason hold estimating functionals type delyon portier main result let support quantity estimated actually estimator modified way leading error term expansion vanishes asymptotically see remark details define factor estimator variance corrected estimator ibc state main result convergences ibc define nikolski class functions regularity constant set bounded times differentiable functions whose derivatives order satisfy stands euclidean norm natural integer careful equal say kernel order soon bounded satisfies notation xldd following assumptions needed show first result discussed statement support compact set integer variable bounded density rth order derivatives bounded every integral approximation kernel smoothing table best acceleration convergence rate theorem best rate acceleration obtained equation equation kernel order moreover exists every exp addition symmetric next theorem proved section theorem assumptions following estimates ibc valid sums inside tend zero remark assumption smoothness crucial guarantee rate faster root theorem one hand one needs obtain rate equation hand suffices get rate equation otherwise exist bounds theorem phenomenon often referred curse dimensionality equation best choice depends balances two three four terms letting one smaller precise rate acceleration situation given table many semiparametric problems see section estimator suboptimal respect density estimation problem see indeed achieve optimal rates density estimation one would need take would even prevent theorem practical bandwidth selection proposed section remark assumption prevents bias problems estimation may occur borders indeed jumps boundary estimate delyon portier would asymptotically biased rates provided theorem would hold get rid problem one knew support one could correct hand estimator instance might use beta kernels detailed remark assumption basically says separated exponential bound kernel assumption guarantees estimated uniformly see helps control random denominators expression ibc context procedures integral approximation assumptions restrictive one draw distribution smooth enough whose support contains integration domain remark use estimators ibc justified simplification involve proofs also leads better convergence rates consider term proof equation theorem replacing estimator classical one remains degenerate nonzero diagonal terms possible show terms leading terms resulting expansion imply rate convergence order larger rate found ibc however concerning estimator necessary get estimator indeed distance ordinary one change would made difference order side already appears side remark function class contains two interesting sets functions provide different rates convergence theorem first constant bounded support second support convex body compact convex set interior constant inside support indicator ball exists min see lemma section sum two nikolski functions still nikolski assumptions theorem valid wide range integrand moreover note loss smoothness boundary support involves loss rates convergence precisely whatever smoothness degree inside support continuity fails boundary nikolski regularity would therefore rates acceleration theorem could exceed section study example show central limit theorem rate remark symmetry assumption actually superfluous simplifies proof case distinguish convolution convolution integral approximation kernel smoothing generalization theorem view intriguing convergence rates stated theorem one may curious know behavior estimator estimating general functionals form following approach previously estimator consider ibt turns given case rates faster root functionals wide range bandwidth ibt converges normal distribution view negative aspect result respect statement theorem provide informal calculation asymptotic law ibt require hold latter guarantees faster root rates equation bounded uniformly derivative using taylor expansion respect second coordinate derivative respect second coordinate denoted ibt treated standard techniques kernel estimation see equations details gives probability going depend write ibt delyon portier nikolski applying theorem gives consequence ibt variance degenerate equivalent want true reasonably large class distribution functions would imply solutions form central limit theorem previous section derived upper bounds convergence rates probability fairly general conditions section little specific regularity able describe precisely asymptotic distribution ibc actually approach decompose latter quantity sum statistic plus martingale respect filtration plus bias term see beginning section definitions existing results asymptotic behavior completely degenerate martingales help derive asymptotic distribution shall consider two cases first present case smooth enough dominant term second study example continuous boundary support consequence dominant term situation less interesting since choice bias term leads asymptotic decomposition see remark smooth case smooth case corresponds situations functions smooth enough highlighted assumptions bandwidth next theorem theorem assumptions random variable ibc asymptotically normally distributed variance given integral approximation kernel smoothing assumptions bandwidth satisfied optimal bandwidths displayed table fact presentation issue indeed chosen make bias term vanish optimal bandwidth balances bias variance excluded could proceeded way around stating ibc limiting distribution theorem provided min one verify holds true optimal bandwidth given first line table equation example interested case sufficiently regular longer negligible respect min occurs whenever case variance hard compute since depends behavior therefore rate convergence kernel regularization hence precise description provided considering usual regularity classes example nikolski sobolev since provide bounds rate kernel regularization reason consider particular case function nikolski inside vanishes outside typical functions mind one jump boundary support lemma informs functions nikoslki regularity compact define min unit normal outer vector point need following assumption place support convex body boundary theorem assumptions random variable ibc asymptotically normally distributed variance given stands hausdorff measure application nonparametric regression equation applications nonparametric regression random design let delyon portier sequence real random variables mean unit variance independent sequence unknown functions let compact set hilbert space functions let extended outside compact support inner product regression function given note belongs given basis coordinate basis among typical applications mention fourier coefficients estimation either nonparametric estimation see section location parameter estimation see also mention link estimation index single index model see estimation linear functionals typical semiparametric problem sense requires nonparametric estimation density first step use order estimate real parameter best knowledge case regression unknown random design estimators achieve root consistency provided yet see reference therein approach based kernel estimates density plugged classical empirical estimator quantity define estimator derive asymptotic use model get decomposition roughly speaking theorem provides negligible respect result therefore limiting distribution carries weak convergence obtained making full use independence order integral approximation kernel smoothing achieve program assumption needed support compact set following theorem proved section theorem assumptions random variable asymptotically normally distributed variance var remark let compare appealing estimator requires knowledge first signal observed without noise goes probability whereas asymptotically normal secondly noise observed signal meaning comparison made regarding asymptotic variances since var asymptotically efficient plug nonparametric estimator use directly remark set reflects domain studied obviously dense stable estimation nevertheless could happen vanishes point taken account framework situations one may adapt estimation sample ignoring design points estimated density takes small values estimator might replaced certainly depend method often referred trimming employed guarantees computational stability well theoretical properties even approach feasible seems far beyond scope article delyon portier simulations section provide insights implementation practical behavior integral approximation procedure particular propose adaptive procedure selects bandwidth kernel smoothing theoretical study highlighted estimators suffers curse dimensionality see remark simulation results confirm estimation accuracy methods diminishes dimension increases dimension procedure outperforms far method moderate sample size dimension method still realizes significant improvement method simulations conducted fairly general design distributions necessarily satisfy assumption equation kernel choice whole simulation study estimator density design based kernel volume unit ball dimension kernel radial order bandwidth choice one may follow select optimal bandwidth method requires optimize asymptotic equivalent mse respect section highlighted limiting distribution mse depends heavily degree smoothness practice regularity often unknown result prefer type strategy idea pick value gives best result estimation integral test function looks like known choose test function simply epanechnikov kernel integral approximation kernel smoothing since know take value estimate closest actually two values one one smoothing parameter chosen using rule thumb given mean estimated variances component see section density estimates computed value kernel try use resampling method thinking better adapted specific sample first model model normal distribution sin integral figure shows simulations different values using equations choice second model second model assumptions satisfied since distribution uniform unit cube sin spite fact satisfied good results still possible cancels boundary cube choice used equation delyon portier figure boxplot based estimates ibc method noted imc first model different values important constrain function support cube way remove boundary terms choosing xij could done way around use simulate uniformly extra points distance less cube order cover support figure shows results simulations different values using equations choice proofs notation euclidean norm norm supremum norm respectively denoted introduce kij integral approximation kernel smoothing figure boxplot based estimates ibc method noted imc second model different values fbi vbi kij kij fbi function define put proof theorem start showing follow straightforwardly delyon portier proof following development reminiscent taylor expansion fbi fbi fbi fbi fbi allows expand estimator sum many terms density estimate fbi moved numerator exception fourth one show last term goes quickly linearised terms messy correct bound obtained expanding also fbi expressions order sort terms borrow vial trick making appear degenerate development inserting right quantity explicitly recalling vbi ibc fbi obtain ibc underbrace terms deliberately introduced removed fbi fbi vbi fbi fbi appears centering term shall compute bounds term separately step note uij uij uij uij integral approximation kernel smoothing uij kij degenerate due fact uij uij terms sum orthogonal norm smaller kuij hence equation lemma step consequence equation lemma assumption step rearrange function chr constant last inequality follows equation lemma chr spliting mean variance first term get var conclude equations lemma easy exercise show nikolski regularity min step first express set fbi vbi delyon portier rewrite consider sequence real numbers set applying kij fixed get kij kik kij going calculate using inequality theorem moment inequalities stated lemma particular var consider function define integral approximation kernel smoothing copy independent sample stein inequality remember order noting terms first sum orthogonal independence conditionally obtain equation terms second sum orthogonal whenever values different get first developing using independent copy obtain delyon portier equation equation get defined lemma bringing everything together nhd holds step start lower bound fbi proving existence notice fbi fbi nhd due almost sure uniform convergence theorem probability large enough inf sup since assumption nhd follows compute expectation restricted real number latter inequality exists constant depend inequality integral approximation kernel smoothing applying fact real number obtain using nhd goes infinity hand using equation putting together particular markov inequality proves boundedness probability step following since show convergence probability side term step indeed rosenthal martingale see delyon portier latter inequality due equation hence conclude step putting together steps taking account concerning obtain proof use shorter expansion leads actually much simpler proof fbi fbi fbi fbi fbi fbi terms already treated steps proof term bounded exactly use instead obtain get proofs theorems let define cij integral approximation kernel smoothing cjk ajk bjk ajk bjk ujk ujk ujk ujk ujk defined beginning step proofs theorems rely following lemma turns theorem suitable way weak convergence issues lemma assumptions theorem ibc moreover proof using decomposition since ibc already shown exactly steps proof theorem definition ajk completely degenerate near completely degenerate ajk appears good centering term order moments quantity order hence shown completes first part proof obtain bounds probability use step proof theorem compute norm follows kmn constant last inequality obtained using equation term right equation lemma term right remark assumption theorem one may show nhd ibc delyon portier comes remainder term corresponds diagonal term term equals plus consequence leading term decomposition constant proof theorem lemma assumptions derive limiting distribution apply theorem quoted theorem cjk ckj cjk ajk bjk defined beginning section asymptotic variance limit quantity asymptotically equivalent compute easily introduce function krh first use algebra obtain formula follows integral approximation kernel smoothing integrates hard see last two terms previous equation negligible computation consequence limit first equality follows change variables last representation follows lebesgue dominated theorem following steps previously obtain similar expression get remains check conditions theorem clearly computation provides obtain similarly implies conditions theorem proof theorem lemma exists min apply lemma assumption obtain since sum independent variables apply central limit theorem checking lindeberg condition see chapter compute asymptotic variance defined limit var one hand equations constant consequence get var delyon portier hand every stands complement set nikolski regularity min inside use equation lemma show norm side term order hmin clearly since min var remains apply lemma derive stated limit proof theorem equation interested asymptotic law fbi fbi lemma side term goes probability term use decomposition fbi define generated set random variables get one fbi fbi inf integral approximation kernel smoothing term left since support use large enough bounded side term follows fbi fbi using equation lemma provide bound fbi therefore shown probability since remains note sequence uniformly integrable apply lebesgue domination theorem get conclude apply central limit theorem statement follows lemmas inequalities lemma function recall assumptions holds kfh positive constant depends proof start proving assuming holds mean using fubini theorem hence delyon portier gives turn use taylor formula lagrange remainder applied order equal largest integer smaller first term polynomial vanish insertion orthogonal first polynomial degree second term bounded hence generalized minkowski inequality page measurable function integral approximation kernel smoothing implies concerning use get latter bounded constant times following lemma gives bounds conditional moments useful proof theorem lemma let kij proof first equation trivial second equation triangular inequality jensen inequality provide third one derived inequality helps bound moments estimators proof refer original paper also theorem inequality let sequence independent copy symmetric function variables var delyon portier measure results lemma let suppose support convex body exists min proof dist called quermassintegrale minkowski dist stands euclidean distance last inequality follows fact steiner formula stated instance theorem page lemma assumption compact set boundary continuous lim min stands hausdorff measure normal outer vector point proof let start estimate integral simpler dependency define function dist integral approximation kernel smoothing function neighborhood gradient normal inner vector since using local parametrization reduced case piece hyperplane actually depends smaller constant related curvature hence integration domain band width hence bounded since second term integral contribution limit negligible suffices prove lim setting latter equality rewritten lim delyon portier proposition page integrable function lipschitz essinf hence obtain letting get lim write zzz min weak convergence degenerate theorem hall let symmetric assume asymptotically normally distributed zero mean variance given acknowledgements authors would like thank vial helpful comments advice latter version article research supported fonds recherche scientifique fnrs integral approximation kernel smoothing references bickel klaassen ritov wellner efficient adaptive estimation semiparametric models johns hopkins series mathematical sciences baltimore johns hopkins univ press boucheron lugosi bousquet concentration inequalities advanced lectures machine learning lecture notes computer science berlin springer chen beta kernel estimators density functions comput statist data anal delecroix hristache patilea semiparametric estimation regression statist plann inference devroye wagner strong uniform consistency kernel density estimates multivariate anal efron stein jackknife estimate variance ann statist evans gariepy measure theory fine properties functions studies advanced mathematics boca raton crc press evans swartz approximating integrals via monte carlo deterministic methods oxford statistical science series oxford oxford univ press federer geometric measure theory die grundlehren der mathematischen wissenschaften new york springer folland real analysis modern techniques applications pure applied mathematics new york new york wiley gamboa loubes maza estimation shifts electron stat hall central limit theorem integrated square error multivariate nonparametric density estimators multivariate anal hall heyde martingale limit theory application probability mathematical statistics new york academic press applied nonparametric regression econometric society monographs cambridge cambridge univ press marron tsybakov bandwidth choice average derivative estimation amer statist assoc stoker investigating smooth multiple regression method average derivatives amer statist assoc jones simple boundary correction kernel density estimation stat comput berger adaptive importance sampling monte carlo integration stat comput simul robinson semiparametric regression econometrica silverman density estimation statistics data analysis monographs statistics applied probability london chapman hall stone adaptive maximum likelihood estimators location parameter ann statist delyon portier stone optimal rates convergence nonparametric estimators ann statist tsybakov introduction nonparametric estimation springer series statistics new york springer vial deux contributions thesis univ rennes zhang nonparametric importance sampling amer statist assoc received september revised march
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deterministic generalized framework unsupervised learning restricted boltzmann machines eric tramel marylou andre manoel francesco caltagirone florent oct dated october restricted boltzmann machines rbms commonly used building blocks deep architectures neural architectures work derive deterministic framework training evaluation use rbms based upon tap approximation systems weak interactions coming theory tap approach extensively studied fullyvisible binary spin systems construction generalized models well arbitrarily distributed spin systems bounded support numerical experiments demonstrate effective deterministic training proposed models able show interesting features unsupervised learning could directly observed sampling additionally demonstrate utilize framework leveraging trained rbms joint priors denoising problems pacs numbers introduction past decade witnessed groundswell research machine learning bolstered deep learning revolution resurgence neural networks since inception researchers identified deep connection neural networks statistical mechanics perhaps unsupervised neural models studied lens statistical physics hopfield model boltzmann machine models proposed connectionist perspective cognitive science studied context emergent representation unsupervised machine learning look hopfield model directly observe contributions physics machine learning cognitive sciences example applying techniques study amit famously able derive memory capacity hopfield model provide concrete understanding dynamics model via study phase transitions fundamental understanding behavior hopfield model provided insight complexities associative memory closely related boltzmann machine undirected stochastic neural network finds physics parallel ising spin glass models specifically model owkin rue pierre paris france laboratoire physique statistique ecole normale rue lhomond paris france neurospin cea france noah ark lab paris huawei technologies france laboratoire physique statistique ecole normale rue lhomond paris france pierre marie curie sorbonne paris france one interested inverse problem learning couplings spins order generate particular set configurations equilibrium process learning couplings training often referred inverse ising problem physics literature however couplings exist pairs spins ising models limited practical application successfully capture correlations might exist set training configurations reason general boltzmann machine introduces set unobserved latent spins effect latent spins abstract correlations within set observed spins optimal training couplings would potentially lead effective general model joint distributions intractability joint latent model confounds practical application general boltzmann machines restricted boltzmann machine rbm special case general boltzmann machine couplings exist latent observed spins bipartite structure key efficient effective training rbms rbms found many applications machine learning problems diverse dimensionality reduction classification collaborative filtering feature learning topic modeling additionally rbms stacked neural networks played historically fundamental role deep network architectures constructions known deep belief networks first truly deep neural architectures leading current explosion activity deep learning access vast training datasets made dispensable certain tasks rbms remain fundamental tool theory unsupervised learning better understanding rbms key future developments emergent machine intelligence date popular effective approaches training rbms centered differing flavors monte carlo sampling cover detail sequel techniques yield trained rbms produce sampled configurations similar target dataset used number applications detailed previously bridge gap understanding rbm learned furthermore understanding modes internal representations rbm frameworks mostly consisted subjective comparisons sampled configurations well subjective analysis couplings often referred receptive fields machine learning literature additionally comparing two trained models even monitoring training one model becomes problematic using investigative tools example annealed techniques provide estimates model large computational cost much lower computational cost used monitor training estimates produced manner inaccurate compared annealed importance sampling ais even ais fail detect model divergence practice present work seek address concerns developing deterministic framework train compare analyze rbms well leverage modeling power inference tasks accomplish via statistical physics techniques use tap formalism theory manner produce model longer refers stochastic model possessing intractable gibbs measure tap machine entirely model operates classical rbm admits deeper introspection via deterministic inference tap machines also naturally handle variables well deep architectures deep boltzmann machines dbms training algorithms mix monte carlo sampling approximation deep tap machine relies entirely tap approximation interpretation tap machine generative probabilistic model deterministic model defining set representational magnetizations given training dataset advantageously learning output computed exactly finite time converging iteration contrast indeterminate stopping criterion monte carlo sampling major distinction tap machine classical rbm true probability density function intractable core tap machine training consists arranging minima solutions proposed free energy maximize correlation solutions dataset experiments demonstrate track growth geometry solutions novel way investigate progress unsupervised learning also show use trained tap machine prior inference tasks paper organized follows sec formally describe classical binary rbm review literature rbm training analysis subsequently sec iii describe proposed modification binary rbm model arbitrary distributions bounded support next sec briefly describe apply perform inference setting spins details approach pedagogically described appendices sec derive tap approximation rbm via expansion gibbs free energy sec detail convert approximation practical training algorithm sec vii conduct series experiments real datasets demonstrating use properties tap machine interpretation provide insight unsupervised learning process additionally show use trained model correction simple example leveraging tap machine inference tasks lastly appendix detail derivations necessary functions restricted boltzmann machines restricted boltzmann machines rbms latentvariable generative models often used context unsupervised learning set weights biases model parameters rbm correspond couplings local fields present system constructs energy function data points follows probability density function binary rbm visible latent variables rbm distribution wij set local potentials set values define biases acting variable wij use notation refer sums entire space possible configurations visible latent variables respectively taken respect parameters model known partition function give representation rbm distribution fig evidenced exact computation normalizing partition function thus probability given data point inaccessible practice sophisticated monte carlo schemes relying importance sampling produce estimates via gradient ascent parameters commonly one calculate gradients training set instead calculates gradients average across often referred gradients given hxi hxi isampled wij hxi hxi isampled hhj hhj isampled fig factor graph representation rbm distribution variables indicated circles latent variables denoted shaded circles shaded rectangles indicated layer partitions within rbm structure factors represented squares right hand side factors representing pairwise relationships variables left hand side factors representing influence localized prior distributions variables bounds partition cost substantial computation running time scale days even weeks thankfully precise estimate normalization unnecessary many rbm applications additionally bipartite structure rbm admits couplings hidden visible variables leveraged construct efficient sampling schemes approach demonstrated contrastive divergence sampling shown sufficient adequate rbm training approach consists sampling chain alternating samples drawn conditional probabilities layer dependent conditional expectations previously sampled layer specifically conditional probabilities hidden visible units factorize sigm wij sigm wij sigm logistic sigmoid function order learn parameters rbm given training dataset one looks maximize following loglikelihood wij refers averages particles sampled model refers clamped expectations values fixed training data samples case expectations involving hidden units unobserved therefore training data originally proposed use configurations sampled however one could also use exact conditional expectations directly calculate clamped averages especially cases sampling conditionals may problematic since number proposed modifications core training scheme described persistent trick takes neatly advantage iterative gradient ascent quickly obtain thermalized markov chains gibbs sampling extra computational cost one step nevertheless probability density function trained rbm typically highly multimodal thus making sampling inexact indeed glassy landscapes mixing becomes slow markov chains become stuck metastable states leading well missed modes highdimensional distribution turn produces high variance estimates means correlations accurate sampling achieved using parallel tempering particles swapped multiple markov chains running differing temperatures approach however requires additional computational burden running chains also requires tuning number chains temperatures run accurate inference rbms costly would seem usefulness limited learning rbm via gradient ascent dependent upon inference difficulty training generative model accuracy compounded however rbms proven useful many applications sampling generative model unneeded instance rbms used unsupervised feature extraction feedforward networks rbms also used tasks image label recovery collaborative filtering reconstructing missing data single step truth training algorithm popularized rbms first binary units gaussian units finally arbitrary units use thermalized samples evaluate means correlations instead focuses region configuration space nearest training dataset using short markov chains starting training data points get fast low variance estimates moments however prone learn spurious minima configuration space far data explore region training also systematically increase true likelihood training data however training strategy found efficient applications mentioned consistently remain close dataset configuration space one finds falls short applications require mcmc sampling trained rbm represents fundamental mismatch training application rbm order address theses shortcomings approaches turn attention deterministic approximations rbm tap approximation disordered systems relies deterministic inference approximated magnetizations one obtain estimators kinds observables starting free energy tap derived small weight expansion variational approach considered extension nmf method previous works attempted make use nmf approximation rbm shown negative results tap approximation first considered boltzmann machines context small random models without hidden units recent work approximation extended practical training algorithm binary rbms shown competitive persistent contrastive divergence pcd applied datasets parallel works used tap related bethe approximation perform inference binary boltzmann machines next sections detail rbm model case generalized distributions visible hidden units similar spirit however unlike earlier techniques approach problem estimating normalization rbm model via tools statistical mechanics resulting framework rbm inference training application interest following manner wij piv pjh sum indicates sum visible hidden units model parameters defining local distributions pjh respectively variables variables case see distribution reduces bipartite spin glass model representing local fields acting system spins fields binary spins described specific case simply binary rbm described previous section already considered within extended framework important distinction model evaluate assume binary discrete distribution variables instead allow formulation variables system possess distributions bounded support considering general class models one include wide range different models including hopfield model rbms data sets images genomic data varying distributions hidden visible units distribution visible variables obtained marginalizing latent variables dhj giving dhj ehj wlj take gradients respect model parameters case distribution terms find iii general distributions rbms turn attention case general rbm grbm distributions hidden visible units fixed define distribution generalizations however case gradient respect couplings find wlj function ebh ebh computes conditional expectation knowing value visible units important note term tractable thanks bipartite structure layer rbm contrast terms second terms eqs require knowledge partials log normalization parameter interest however term written exactly explicit calculation normalization intractable rather resorting sampling attempt approximate free energy parametric deterministic way next section discuss belief propagation used estimate conduct inference rbms strong correlations within receptive fields order construct algorithm pdfs realvalued support one requires finite memory description messages examples descriptions given moment matching relaxed appendix following example show arrive approximation continuous messages via smallweight expansion rbm coupling parameters also show approximated free energy pairwise models well demonstrating need distributions bounded support order preserve bounded messages next section building upon derivation consider approximations rbm via hightemperature plefka expansion approximation via belief propagation one method might estimate partition via review appendix pairwise models rbm essentially given factor graph joint statistical model rbm fig algorithm attempts estimate set marginal distributions variable case graphs provides exact calculation marginals application factor graphs containing cycles loopy guaranteed provide accurate estimates marginals however often estimated marginals significant overlap true ones additionally known solutions loopy bethe free energy allows construction approximation applying inverse learning problem one compute gradients bethe free energy terms parameters model allowing gradient ascent training data one significant hurdle application loopy rbm learning variables messages propagated edges factor graph continuous pdfs case discrete variables ising potts spins messages written using magnetizations full discrete pmf respectively binary variables approximations boltzmann machines considered context inference fixed parameters similar study binary rbms conducted loopy important note studies investigated properties boltzmann machines random weights studies permit many analytical tools studying behavior rbm one directly map observations rbm inference practice trained weights may exhibit tap approximation pairwise models one could utilize approach order estimate free energy generalized spin model detailed earlier section approach might desirable practice specifically one wishes solve inverse learning problem estimating model parameters given dataset necessary estimate gradients model parameters model likelihood parameter update using approach detailed sec iii requires one estimate gradients many thousands times systems large size scales quite poorly estimating gradient requires iteration equations messages additionally one must distinguish cavity terms marginal terms final gradients desire estimated using marginal terms alone requiring iteration set messages extremely costly operation instead one turn approach writing free energy stationary points terms marginals alone done including certain correction terms specified degree weights context rbms approaches proposed order naive order using tap equations introducing additional correction term case grbm arbitrary distributions unit however must tap approximation terms parameters distributions well approximate marginalized distribution site first two moments task turns closely related tap approach matrix factorization possible derive stationarity conditions inferred marginals messages directly taylor expansion rather focus free energy directly provide gradients require training grbm parameters via expansion present lastly point tap free energy secondorder tap term depends statistical properties weight distribution derivation presented assumes independent identically distributed weights scaling assumption simplification practice weight distribution known priori distribution depends training data changes throughout learning process according training adaptive tap adatap formalism attempts correct assumption allowing one directly compute correct correction term realization matrix without hypothesis entries distributed although algorithm principled approach computational complexity almost rules implementation moreover practical learning experiments indicate training using adatap differ significantly tap assuming weights detailed discussion computational complexity learning performance described appendix assume entries scale sites widely connected order size system apply tap approximation high temperature expansion gibbs free energy case boltzmann distribution global minima gibbs free energy value equilibrium matches helmholtz free energy derive twovariable parameterization gibbs free energy derived via legendre transform additionally show gibbs free energy variational attains helmholtz free energy minima clarity notation make derivation terms pairwise interacting hamiltonian without enforcing specific structure couplings bipartite structure rbm reintroduced section first introduce inverse temperature term facilitate expansion wij see fields disappear recover true helmholtz free energy additionally note following identities augmented system namely hxi average augmented system given auxiliary fields since partial derivatives helmholtz free energy generate cumulants boltzmann distribution shown hessian simply covariance matrix subsequently positive hence convex function terms convexity shown true log partitions exponential family distributions take legendre transform introducing conjugate variables derivation tap free energy system effect two auxiliary fields wish derive gibbs free energy system terms first two moments marginal distributions site accomplish proceed first defining augmented sup define solution auxiliary fields defined make explicit dependence auxiliary field solutions values conjugate variables looking stationary points auxiliary fields find hxi hxi implication identities valid unless meets constraints first second moments marginal distributions augmented system wish show correspondence helmholtz free energy unique minimum first let look stationary points respect parameters take derivative gibbs free energy decomposed sum independent terms specifically take expansion terms careful application chain rule find carrying similar computation derivative respect provides shows solution gibbs free energy must satisfy true event solutions auxiliary fields truly looking inverse legendre transform gibbs free energy find inf implies minimum gibbs free energy equivalent helmholtz free energy holds since convex legendre transform convex function convex since gibbs free energy therefore possess single solution minimum must satisfy therefore must finally rewrite gibbs free energy defined function moments parameterized grbm parameters normalization boltzmann distrie temperature bution defined find first term expansion directly lagrange multipliers given functions temperature order make clear order apply later exact form gibbs free energy intractable original free energy apply taylor expansion order generate approximate gibbs free energy make expansion limit infinite temperature interactions sites vanish system described terms individual sites relationship system average local potentials allowing dxi recognize last term simply normalization distribution whose moments defined define tap free energy writing remainder expansions terms specific case wij wij still need define extremal values define gibbs free energy taking stationarity expanded gibbs free energy respect find wij make definitions convenience direct allusion definitions cavity sums inference given eqs conversely deriving stationarity conditions auxiliary fields obtain equations show tap free energy valid following hold substituting values closes free energy marginal distribution moments completes derivation free energy approximation defined elements versus values required solutions tap free energy given tap free energy valid equations met stationary points thus certain set physical meaning additionally know minima exact gibbs free energy correspondence original exact helmholtz free energy exact gibbs free energy terms moments convex exponential family tap free energy possess multiple stationary points whose number increases rapidly grows later sec vii show grbm training progresses number identified tap solutions explained due variance weights growing training fixed use practical grbm implementation variance weights serves effective inverse temperature increasing magnitude identical effect system cooling increases additionally gibbs free energy correspondence helmholtz free energy minimum necessarily true tap free energy approximate nature expansion removes correspondence thus may possible ascertain accurate estimate helmholtz free energy single set inferred shown fig case estimate implies gibbs free energy true one attempt find minima order find accurate estimate foundational principle variational approaches however extra expansion term tap free energy improve accuracy modeling provide lower bound estimate tap free energy could instead one might attempt obtain estimate helmholtz free energy utilizing either subset equilibrium solutions tap free energy since manner might distinguish equilibrium moments proximity unknown averaging theetap free energy across solutions serve simple estimator denoted weighting introduced average correcting helmholtz free energy estimate low temperature removing exponential number solutions weights approach proportional exponents solution tap free energy placing much stronger emphasis solutions however approach general setting expect large deviations expectations derived model additionally weighting scheme shown across entire set solutions particular random model case interested solution space centered particular dataset wish model since fig cartoon description estimating helmholtz free energy dotted via gibbs blue dash tap red free energies example convex gibbs free energy exists one unique minimum moments gibbs free energy matches range tap free energies gray box gives boundary location averaging tap free energies provides estimate solutions computed iterating tap selfconsistency equations easily probe region initializing iteration according training data subsequently encounter band solutions must weight instead obtain set solutions small region support tap free energy due uniformity solutions averaging across solutions seems best approach terms efficiency subsequent section explore properties numerically trained rbms rbms tap machines utilize tap inference sec need write tap free energy terms variables rbm clarify bipartite structure grbm rewrite tap free energy terms hidden visible variables fixed temperature frbm ziv biv avi biv avi avi avi cvi zjh bjh ahj bjh ahj ahj ahj chj wij avi ahj cvi chj means variances visible hidden variables respectively sec solutions tap grbm free energy found iteration shown alg bears much resemblance amp iteration derived context compressed sensing matrix factorization note rather updating entire system time step fixing one side time effect stabilizing fixedpoint iteration clarity alg written single initialization visible marginals however noted sec exist large number initializationdependent solutions tap free energy thus order capture plurality modes present tap free energy landscape one run inference independently many different initializations use case grbm requires train grbm tightly data space data imputation makes sense fix initializations inference points drawn dataset order train grbm holistically structured random initializations help probe modes outside data space set tap solutions fixed grbm parameters tapapproximated written zjh wij frbm avk cvk ahk chk zjh normalization conditional expectation since averaging samples gradients grbm model parameters given wij wij cvi chj wij zjh ahj ziv avi algorithm tap inference grbms input initialize repeat hidden side updates wij fah fch visible side updates wij fav fcv convergence presented gradients make point set data samples set tap solutions different cardinality example one might employ strategy training set data samples used gradient calculation might order however depending application grbm one might desire probe large number tap solutions order accurate picture representations learned grbm case one might start large number initializations resulting large number unique tap solutions contrary one might start number initializations equal number unique solutions might especially early training number hidden units small using gradients simple gradient ascent fixed monotonically decreasing used update grbm parameters present final grbm training algorithm alg besides considering units another natural extension traditional rbms consider additional hidden layers deep boltzmann machines dbms possible define train deep tap machines well probabilistic dbms substantially harder train rbms clamped terms gradient updates become intractable depth interestingly training algorithms retain monte carlo evaluation intractable terms introducing approximation terms deep tap machines consistently utilize tap equations explicit definition training algorithm fully described appendix wij wij wij normalized tap algorithm grbm training input initialize wij repeat size alg training epochs end experiments datasets mnist mnist handwritten digit dataset consists training testing set samples respectively data samples pixel grayscale images normalize dynamic range images centered crops digits roughly balanced proportion construct two separate versions mnist dataset first refer applies thresholding pixel values set others second simply refers normalized dataset introduced cbcl cbcl face database consists face grayscale pixel images experiments utilize face images database contains training testing samples face images experiments normalize samples dynamic range cbcl prior vis gauss prior hid table parameter settings grbm training training epochs cbcl normalized tap vii normalized tap training epochs fig training performance epochs tested datasets varying numbers hidden units performance measured terms normalized tap estimate computed training data samples tap free energy estimated using unique tap solutions thermalized initial conditions drawn data samples thermalization determined convergence magnetizations difference mse iterations solid lines indicate average normalized tap tested training samples shaded regions indicate standard error learning dynamics investigate behavior grbm course learning procedure looking metrics interest training dataset tap free energy number discovered tap solutions note metrics unique model grbm active features localized features least active features cbcl spread features active features least active features spread features localized features active features spread features localized features least active features fig subsets final receptive fields columns obtained tap training grbm models varying numbers hidden units receptive fields dark blue yellow mapped respectively green indicates value receptive fields ranked according two criteria first spread conversely localization measured receptive field second activation level measured mean activation receptive field corresponding hidden unit averaged across training dataset shown indeed maximize tap case binary rbms specific construction entirely independent tap model grbm thus hard say grbm better general case present present comparisons tap grbms varying complexity trained fixed settings indicated table fig see comparison tap function training epochs binary rbms consisting differing numbers hidden units performed training data define one epoch single pass training data every example presented gradient ascent specifics particular experiment given caption note equal comparison across varying model complexity normalized number visible hidden units present model way observe tap gives measure concentration representational power encapsulated unit model increasing values normalized tap indicate evaluated training samples becoming likely given state grbm model parameters observed level complexity tap data rapidly increases values quickly adjust random initializations receptive fields correlated training data however across tested models epoch rate increase tap tapers constant rate improvement reference also show subset trained receptive fields rows tested experiments since full set receptive fields would large display attempt show representative samples fig looking extreme samples terms spatial activity training set observe trained grbms case able learn localized stroke features commonly observed literature binary rbms trained mnist dataset interesting note even case using novel implementation truncated visible units able observe similar learned features case take empirical indication proposed framework grbm learning truly learning correlations present dataset intended finally see feature localization increase number hidden units date understanding rbm learns unlabelled data mostly purely subjective exercise studying receptive fields shown fig however interpretation grbm tap machine provide novel insight nature dynamics grbm learning via stationary points tap free energy detail next section probing grbm given deterministic nature tap framework possible investigate structure modes given set grbm parameters produces free energy landscape understanding nature concentration modes gives intuition representational power grbm cbcl fig distribution free energy estimates tap solutions function training epochs three different datasets case two mnist experiments number hidden units samples drawn training data used initial conditions cbcl training samples used top row tap free energy unique tap solutions transparent blue dots helmholtz free energy estimate via uniform averaging red line number unique tap solutions also given green line bottom row detail tap free energy distributions slices training histograms given bars kernel density estimates tap free energy distribution given curves date observing modes given grbm model could approached via sampling given enough sampling chains diverse set initial conditions thermalizing chains produces set samples one could attempt derive statistics concentrations samples space attempt pinpoint likely modes model however number required chains resolve features increases dimensionality space number potential modes might exist space numerical evaluation carry would impractical sampling techniques models rbm allow directly obtain modes model running inference solve direct problem given diverse set initial conditions given training dataset running tap provides deterministic mapping initial conditions drawn data nearest solution tap free energy initial point drawn dataset solution interpreted rbm bestmatching internal representation data point large number structurally diverse data points map single solution may indicator grbm parameters sufficient model diverse nature data perhaps changes model parameters required conversely number solutions explodes roughly equivalent number initial data points indicates potential phase specific rbm perhaps memorizing original data samples training additionally phase large set tap solutions may replete spurious solutions convey little structural information dataset case model may need tuned order ensure model possess meaningful generalization data space observe effects obtain subset tap solutions initializing tap iteration initial conditions drawn data set running iteration convergence counting unique tap solutions present measures solutions fig count number unique tap solutions well distribution tap free energy solutions across training epochs common features across tested datasets first early phase training shows marked increase tap free energy gradually declines training continues comparing point inflection tap free energy normalized tap loglikelihood shown fig shows early phase grbm training dominated reinforcement empirical moments training data grbm model correlations playing small role gradient makes sense random initialization implies hidden units almost independent training data thus tap solutions early stage learning driven correlated local potentials hidden visible variables effect influence tap free energy landscape possesses modes data space fig shows clearly number tap solutions starts steadily increases ing positive dominant grbm parameters appear minimize tap free energy would expect however tap solutions appear data model terms gradient become balanced tap free energy minimized point inflection see leveling normalized tap second observe bands tap solutions feature especially pronounced case experiment training epochs exist two significant modes free energy distribution tap solutions see effect clearly histograms shown bottom row fig case experiment see free energy distributions exhibit tight banding show presence solutions persist across training main feature across experiments structure free energy distribution finally note case empirically observe explosion tap solutions potential indicator phase since proportion unique tap solutions initial data points remains less order investigate whether modes tap free energy distributions randomly assigned configuration space exist separate continuous partitions configuration space need look proximity solutions configuration space space observed ambient dimensionality project configuration space embedding fig utilize well known isomap algorithm calculating manifold approximately preserves local neighborhoods present original space using visualization observe training progresses assignment high low free energy tap solutions appear random nature seems inherent structure solutions location configuration space additionally fig see progression tap solutions many spread across configuration space interesting note solutions start highly correlated state proceed diversify also observe tap solutions respect initializations produced shown fig charts use similar approach fig mapping data points well tap magnetizations embedding using isomap allows see approximate way tap solutions distribute data space also show number tap solutions grows many training maintain spread distribution data space demonstrates training procedure altering parameters model place tap solutions within dense regions data space sake clarity included lines indicating attribution initial data point resultant tap solution however training progresses one sees tap solutions act attractors data space clustering together data points tap machine recognizes similar inference denoising serving prior inference one particular use case tap machine interpretation grbm simple demonstration turn common signal processing task denoising specifically given planted signal one observes set noisy observations measures true signal corrupted stochastic process denoising tasks ubiquitous signal processing analog level additive shot noise level digital communications binary symmetric erasure channels goal task produce accurate estimate unknown signal analog case may measure mse estimate true signal binary case may measure accuracy counting number incorrect estimates function binary confusion matrix matthews correlation coefficient mcc fixed set observations channel parameters assume original signal drawn unknown intractable generating distribution construct accurate tractable approximate priors accurately construct estimate original signal words know structure content unknown signal priori closer estimate often case image denoising statistics gathered transform coefficients particular images classes heuristic denoising approaches designed accordingly derivation denoising algorithms works well practice owing generality specific priori information original signal required beyond signal class natural images human speech radar return timings however meaningful features must assumed investigated practitioners successful inference take place epoch isomap dim epoch isomap dim epoch isomap dim epoch isomap dim isomap dim epoch epoch isomap dim isomap dim epoch isomap dim isomap dim epoch isomap dim isomap dim epoch isomap dim isomap dim epoch isomap dim isomap dim epoch isomap dim isomap dim epoch isomap dim isomap dim epoch isomap dim isomap dim epoch isomap dim isomap dim epoch isomap dim isomap dim isomap dim epoch high isomap dim relative tap isomap dim epoch isomap dim isomap dim epoch isomap dim isomap dim epoch isomap dim isomap dim low isomap dim isomap dim isomap dim isomap dim fig isomap visualization tap solutions training epochs tap solutions mapped embedding via isomap transform fitted tap solutions epoch embedding performed hidden visible inferred expectations color mapping corresponds tap free energy values solution range colors normalized minimum maximum free energies solution training epoch epoch epoch class digits class digits class digits isomap dim class digits class digits class digits isomap dim class digits class digits class digits isomap dim class digits class digits class digits isomap dim isomap dim epoch epoch class digits class digits class digits isomap dim epoch isomap dim isomap dim isomap dim epoch class digits class digits class digits isomap dim class digits class digits class digits isomap dim class digits class digits class digits isomap dim isomap dim cbcl epoch epoch data points isomap dim epoch isomap dim isomap dim isomap dim epoch isomap dim epoch isomap dim isomap dim isomap dim epoch data points isomap dim data points data points isomap dim isomap dim fig comparison initial conditions tap equilibration colored dots compared converged tap solutions black dots tested datasets different stages training dataset isomap embedding calculated initialization data subsequently magnetizations tap solutions embedded space case initial variances set also random selection tap solution magnetizations chosen provide context representations rbm learning digits corresponding classes drawn first training samples binarized mnist dataset initializations reduced set labels used readability initializations used however initializations binarized cbcl available training face images used initializations denoising binary symmetric channel likelihood observation binary denoising problems assume binary symmetric channel bsc defined following mann ner given binary signal observe signal independent bit flips occurring probability gives following shown equivalent representation boltzmann distribution given prior distribution posterior distribution given bayes rule sigm ope assuming factorized mxi might empirical averages obtained available training data posterior factorizes construct pointwise estimator ope average hxi binary problem thus ope site given mcc given dataset ope gives performance using pointwise statistics dataset namely empirical estimates magnetizations see ope either returns observations case prior magnetizations case case complete information loss worst case performance bounded according deviation dataset mean present performance ope fig dataset makes ope valuable baseline comparison sanity check grbm approximation grbm model takes account pointwise pairwise relationships data properly trained grbm provide estimates least good ope algorithm represents different heuristic approach problem case noisy measurements compared set exemplars training dataset according distance metric mse correlation one finds exemplars minimal distance noisy observations serve basis recovering original binary signal one use arbitrary approach fusing exemplars together final estimate simplest case would simple average case performance using averaging bounded empirical magnetizations limit estimate simply nearest exemplar hard show limiting performance approach dependent distances chosen metric exemplars observations well interplay noise channel distance metric however seen directly approach approach yield true bit flip probability fig average denoising performance reconstruction errors probability bit flipped denoising via inference rbm denoised varying numbers hidden units also shown baseline comparisons ope given empirical factorized magnetizations site dashed black matching training set solid black experiments run data samples drawn test set compared using mcc binary estimates obtained ope tap inferred estimates rounding resulting magnetizations signal unless true signal contained within training data show performance fig advantage approach successfully regularizes noise nearest exemplar always least marginally correlated original signal distance metric additionally see performs better ope regime explained since think approach implicitly though indirectly taking account higherorder correlations dataset returning data exemplars estimates trivially posses arbitrarily complex structure unknown signal using grbm hope capture best points approaches first hope perfectly estimate original signal case second hope leverage pairwise correlations present dataset returning estimates retain structure data even grbm denoising bsc longer factorized posterior instead grbm likelihood given summed hidden units using definition binary prior given appendix wij ope since grbm bsc channel likelihood written exponential family distributions wij finding averages hxi model simply consists running inference alg modified visible binary prior one heuristic caveat approach must take account nature tap free energy since must initialize somewhere resulting inference estimate dependent upon initialization initialize inference ope result see highest probability configuration becomes observations limit able obtain true signal ope especially since initialize within well potential shown binary rbms trained varying numbers hidden units fig every case true signal recovered case see tap inference binary rbm always outperforms ope additionally see cases performance closely mirrors limit see result tap inference essentially uncorrelated original signal case extra potential present bias inference resulting estimate simply arbitrary solution tap free energy closely mirrors exemplar selection mcc curves two approaches similar case see effect occurs essentially low values tap inference binary rbm able accurately identify original signal however certain point owing increased number solutions tap free energy exist many undesirable minima around noisy solutions leading poor denoising estimates one observe subjectively fig case tap inference results either nearly localized ones would seem indicate landscape tap free energy around initializations becoming unstable density solutions increases around additionally since tap free energy landscape probed using data points training clustering solutions around noisy samples remains ambiguous augmenting initializations used calculating tap solutions gradient estimate noisy data samples could help alleviate problem regularize tap free energy landscape space noisy data samples fig subjective comparison denoising estimates single digit image ope rbm approaches inferred posterior averages hxi shown rather final configurations white black represent values respectively tested value noise realization used method increases tap inference rbm provides estimates still possess digit structure case tap inference gets caught spurious undesirable minima increases viii discussion paper proposed novel interpretation rbm within fully tractable deterministic framework learning inference via tap approximation deterministic construction allows novel tools scoring unsupervised models investigation memory trained models well allowing efficient use structured joint priors inverse problems deterministic methods based nmf rbm training shown inferior level approximation accuracy afforded tap finally makes deterministic approach rbms effective shown case binary rbms additionally construction generalized distribution hidden visible units unique work works propose unique training methods models changing distribution visible units example seen modified hamiltonians used data construction allows consider binary sparse datasets within framework additionally one also consider architectures changing distributions imposed hidden unit present experiments using binary hidden units one could also use proposed framework hidden units thus mimicking hopfield network also sparse gaussbernoulli distributed hidden units could mimic functionality proposed rbm left investigations works topic proposed framework also offers possibility explore statistical mechanics latent variable models level tap approximation specifically given statistical model weights cavity method replica begin make predictions unsupervised models analytical understanding complexity free energy landscape transitions function model allow richer understanding statistically optimal network construction learning tasks case random networks already progress area shown however similar comprehensive studies conducted learning realistic setting still yet realized finally framework applied deep boltzmann machines minimal alteration also potentially lead richer understanding deep networks role hierarchy regularizing learning problem high dimensionality joint distribution wij sum edges graph case boltzmann machine two variables connected via pairwise factors wij ewij variables also influenced univariate factors written trivially constructed factor graph writing messages variables factors also factors variables since factors degree write messages system variable variable messages dxl wil acknowledgments research funded european research council european union framework programme grant agreement acknowledges funding chaire recherche sur les sciences des fondation cfm pour appendix belief propagation pairwise models order estimate derivatives must first construct algorithm factor graph representation given fig note graph terms variables make explicit distinction latent visible variables instead treat graph full generality clarify derivation notation graph corresponds following notation represents message variable index variable index refers neighbors variable index except variable index denote neighboring variables share pairwise factor finally messages refer iteration implies successive application convergence set messages set pairs neighboring variables also note inclusion message normalization term ensures messages valid pdfs additionally possible write marginal beliefs variable collecting messages neighbors dxj wij subsequently bethe free energy written converged set messages according dxi dxj wij refers normalization set marginal belief site derived unfortunately written computable algorithm due continuous nature pdfs instead must find manner parameterize messages case binary variables message pdf exactly parameterized expectation ever general case formulation make assumption instead turn relaxed described next section assumes parameterization messages appendix pairwise models consider one possible parametric approximation message set via approach also gone number different names parallel approach moment matching essence assuming messages mean variance gaussian assumption approximation arises expansion assuming small weights wij making assumption ultimately able close approximation messages two first moments hxi hxi approximate series dropping terms less approximation justified event weight values satisfy identifying integrals expansion moments see following approximation wil however would like write approximation terms central second moment second approximation neglects terms arrive desired parameterization incoming message marginalization terms message two first central moments exp log wil exi wil wil wil exi wil wil see set closed equations due dependence moments vice versa values moments written function dependent upon form local potentials prior distribution assign variables simply normalization inferred marginal distributions site calculated via functions instead using incoming messages appendix give moment calculations different choices one wants obtain estimate free energy given set parameters possible iterate ideally convergence important note however due potentially loopy nature network well smallweight expansion iteration guaranteed converge additionally retain derivation clear whether one attempt iterate message fully sequential parallel fashion clustering partitioning variables applied determine update order dynamically substitute approximation back get considering marginalization taking place perform second order expansion assuming wil start taking taylor series incoming message marginal negligible weights dxl ewil wil wil dxl wil wil dxl wil wil derivation via small weight expansion approximate bethe free energy additionally write specific form bethe free energy parameterization messages case simply apply small weight expansion bethe free energy smaller prior messages become unbounded fail meaningful probability distributions expansion fails implication message passing utilized contexts exists preferably strong evidence site dxi exi wij wij weights sufficiently small stronger local potential smaller weights favorable model inference observation mirrors made however subsequently using final message definition given authors make observation setting see based expansion binary variables fails converge case without taking form regularization inference fails entirely thus large magnitude couplings wil must backed high degree evidence site property could correcting double counting also written utilized inverse learning problem one must learn couplings given dataset order constrain learning parameters amenable inference one direct manner create probability distributions otherwise unbounded continuous functions via degree site truncation specifically enforce normalization factor restricting support distribution subset case slightly violating enforcing bounded messages bounded condition induce uniform message distribution distribution support strong write messages though violation cause distribution concentrate gaussian distributions slight since boundaries support another approach might implication expansion simply fix hard boundary constraint general messages fact unbounded thus never permitting unbounded messages occur unboundedness direct result form conventional rbm pairwise factor exi wij avenues available address unbounded messages produce meaningful messagepassing generalized rbms let consider cases appendix calculations specific variable messages unbounded given specific distributions variable distribution assume site assigned gaussian prior truncated gaussian units case message reads pairwise models given case message unbounded event weighted sum incoming neighbor variances exceeds inverse variance gaussian prior variance gaussian prior said another way condition telling starts tell variance general truncated gaussian defined following manner mean variance original gaussian prior truncation range defines lower upper bounds truncation cdf normal distribution make things easier later define prior little bit different manner making following definitions writing distribution parameters erf erf erf error function last step folh lows identity erf subscript used indicate case write terms imaginary error function erfi noting erfi erfi use negative variance version truncated gaussian handle special case first detailed sec detail computation partition first two moments distribution calculation moments function provide definitions calculation normalization provide terms necessary computation tap free energy well gradients necessary learning rbm training first calculate normalization terms free parameters consider truncated normalization following product gaussians defined need make note special case thus erf erf erfi erfi since simply truncated gaussian updated parameters first moment given according well known truncated gaussian expectation expectation usually written terms mean variance gaussian distribution case positive mean instead write expectation terms exponential polynomial coefficients note special case next write variance function earlier since specific form truncated gaussian distribution utilize variance formula distribution case modify function special case specifically gradients determine gradients model parameters necessary calculate gradients terms distribution parameters assume boundary terms remain fixed since distributions truncated gaussians treat terms derivatives log normalization truncated gaussian boltzmann measures given quadratic form know hxit relations write necessary derivatives case hxit see difference data moments truncated gaussian distribution next derivatives fixed written terms moments hxit finally ready write gradients loglikelihood used hidden visible updates truncated gaussian distribution given set data indexed number tap solutions indexed gradients visible variable given hxi hxi hxi hai form erf erf rewritten erf multiple difference two boundaries truncated gaussian distribution since wish consider case take taylor expansion centered since value dominates find following approximation works well practice averages tap solutions respectively updates hidden side variables using truncated gaussian distribution following gradients updating parameters haj moments aej wij cej wij defined convenience using gradients update local biasing distributions truncated gaussian variables power series representation error function experiments use similar approximation used variances situation approximation could potentially computationally costly large note used updates variables small value detected truncated units truncated distributed units form distribution mixture delta function extra term controls density thus numerical considerations using truncated gaussian prior comes numerical issues must carefully considered already addressed cases addressed case one see complicates matters observing term occurs first second moment computations magnitude limits become vanishingly small comparison scaled joint mean term see however numerators also zero implies may able find method approximation find estimates moments without numerical precision dividing zero order handle eventuality tion make taylor series expansion term following way first note dirac delta function everywhere else using construction truncation done gaussian mode alone across entire distribution easily write necessary functions distribution terms values already calculated appendix long additionally useful calculations define probability according continue previous appendices write normalization first two moments gaussian product distribution first normalization written simply function truncated gaussian normalization modified next first moment found recalling hxiq consequently relation hxiq thus fat probability calculated according second moment note find fat next turn attention gradients necessary updating parameters training first look derivatives required updates visible units order calculate derivatives split log probability two cases consequently derivatives log probability written following hxit rewritten concise form hpq visible units peq hpq hidden units peq calculated field manner described appendix wij binary units define distribution binary units bernoulli distribution derivative bit complicated use identities additionally must also consider two cases separately thus prob hxib also write boltzmann distribution gives final gradients fct fct fat remain consistent modification moments taken need write gradients starting log partition applying identity used write noting complement support probability making appropriate substitution write derivatives terms take form written appendix subsequently equations next calculate normalization moments distribution normalization subsequently moments sigm sigm logistic sigmoid function subsequently variance binary unit calculated directly next wish define learning gradients write derivatives hxib log normalization hxib resulting gradients distribution terms hxi hai visible units hai hidden units wij appendix adaptative tap performing inference one could employ instead tap variant known adaptive tap adatap gives general accurate results albeit slower iterate briefly investigate performance method binary case adatap algorithm generally presented without distinction visible hidden variables thus write algorithm generic weight matrix bias vector practice defined blocks proposed implementation alg uses recently introduced vector approximate vamp find adatap fixed points convergence quantities subscripts equal identify outputs tap inference algorithm alg compactly defined blocks visible hidden units defined diagonal matrix gives estimate correlation different units must computed step algorithm quantities incorporated training algorithm alg computational burden alg lies matrix inversion needed evaluate needs performed iteration fig right compare time needed perform one iteration algorithms identical experimental conditions larger cost compensated principle accurate inference procedure however seem translate improvements training performance fig left presents minimal test mnist training samples performances reported terms evaluate algorithms different numbers iterations results suggest strategies roughly equivalent except running adatap small number iterations always leads poorer result thus conclude far proposing tractable efficient training algorithm rbms tap inference seems serve purpose appropriately appendix deep boltzmann machines model inference possible define well deep models boltzmann machines considering several stacked hidden layers deep boltzmann machines dbm consist straightforward extension rbms distribution corresponding dbm hidden layers indexed wij wij piv similarly rbms distribution visible variables obtained marginalizing latent variables dhj yielding following dhj wij wij major difference rbms dbms lies complexity evaluating expression whereas rbms features problematic multidimensional integral logpartition last term intractable additional complication carries computation gradients necessary training since datadependent term deriving last term longer tractable stance effective weight matrix dbm hidden layers defined blocks intractability follows fact hidden units neighboring layers connected thus longer conditionally independent interestingly first proposal deal datadependent terms dbms consisted using naive approximation keeping monte carlo based strategy compute gradients deriving work propose instead use tap approximation hence improving nmf approximation avoiding sampling rather complicated rbms thus implementing grbm inference algorithm alg proper weights outputs tap solutions vector components corresponding different units dbm related tap equations follow directly general derivation sec fully connected models however different weight matrix used last term recognize model closely related considered dbm visible units anymore variable fixed clamped values original interaction visible first hidden layer units replaced additional local field equal wij finally simple modification hamiltonian tap equations follow general derivation sec resultant tap solutions depending data points said clamped denoted training algorithm experiments gradients respect model parameters similar rbm ones given however first term analytically computed anymore use clamped tap solutions approximate second term evaluated using tap solutions similarly strategy rbms corresponding expressions gradients ziv avi wij cvi chj input initialize repeat prior updates sigm interaction updates diag zjh algorithm adatap inference rbms convergence wij cvi chj expressions plugged gradient ascent algorithm rbm training algorithm alg nevertheless simple strategy simultaneous training parameters model joint training usually fails magnitude weights deep layers typically remains small model eventually resembles mere rbm several regularizations proposed tackle problem dbm training experiments used greedy layerwise consists computing meaningful initialization weights training rbms performing joint training complete algorithm described alg fig shows evolution tap layer layer dbms trained described algorithm algorithm gdbm training input tpretrain tjoint train pretraining alg tpretrain alg tpretrain end joint training initialize repeat size alg alg wij wij end tjoint train pseudo log likelihood adatap tap adatap adatap adatap tap tap tap training epochs time batchsize fig left evolution along training rbm binary visible units binary hidden units training performed using first images binarized mnist learning rate batches size different curves correspond different strategies estimation likelihood gradients either tap adatap algorithms iterated fixed number times cases damping used methods yield comparable results terms training performance except adatap iterations shows poorer performance right computation time one iteration inference algorithm function batch size time reported seconds identical experimental settings need matrix inversion batch element makes vamp orders magnitude slower tap hidden layers test train normalized tap log likelihood normalized tap log likelihood test train hidden layers training epochs training epochs fig training performances training epochs layer left layer right deep boltzmann machines dbms binarized mnist datasets models pretrained epochs learning rate training performance measured normalized per unit tap test images blue train images orange lecun bengio hinton deep learning nature hopfield neural networks physical systems emergent collective computational abilities proc nat acad sci amit gutfreund sompolinsky 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algorithms greater good mental modeling acceptable symbiosis collaboration tathagata chakraborti subbarao kambhampati jan department computer science arizona state university tempe usa rao abstract effective collaboration humans systems requires effective modeling human loop terms mental state well physical capabilities latter however models also open pathways manipulating exploiting human hopes achieving greater good especially intent values human aligned asymmetrical relationship respect knowledge computation power fact behavior necessarily require malicious intent rather borne cooperative scenarios also beyond simple misinterpretation intents case value alignment problems thus effectively engineered desired techniques already exist pose several unresolved ethical moral questions regards design autonomy paper illustrate issues teaming scenario investigate perceived participants thought experiment promise collaborations systems become integral parts daily life workplace essential components hitherto enterprises effects interaction humans automation ignored terms partnerships affect outcome activity evolve result also terms possibility interactions change design autonomy light traditional view substrate complete autonomy automation facto ever since conception field somewhat evolved late accommodate effective symbiosis humans machines rather replacement former latter one principal end goals design autonomy view fact reflected heavily public stance network world many industry leaders technologies diverse fields manufacturing medical diagnosis legal counseling disaster response military operations others establishment collaborations people systems partnership pai one thematic pillars partnership primary example one grand goals design integrate best worlds comes differing often complementary expertise humans machines order conceive whole bigger sum capabilities either referred augmented bird public discourse integration much discussion around topic augmentation versus replacement unfortunately centered around mitigating concerns massive loss employment account latter topic worthy debate represent true scope collaborations rather foil concerns replacement humans systems key objective overcome human limitations involve helping humans tasks traditionally good incapable performing even augmentation physiological form realize capabilities tom gruber siri put succinctly ted talk tom gruber earlier year every time machine gets smarter get smarter examples include smart assistants personal business use law health care science education assistive robots home help sick elderly autonomous machines complement daily lives note many applications inherently symbiotic thus outside scope eventual replacement perspective research well attitude towards including human loop design autonomy seen significant shift originally often looked upon means punting hard challenges designing autonomous systems introducing human expertise agents decision making process however academic community gradually come terms different roles human play operation aisystem vast challenges research come interactions complement limited capabilities system seen cobots veloso ask humans vicinity access different floors elevator horvitz automated planners old complement expand capabilities human teams christensen mental modeling forms collaboration introduce typical research challenges otherwise absent isolated design autonomy perhaps difficult aspect interacting mans need model beliefs desires intentions preferences expectations human situate interaction context model believe one hallmarks rachael rettner human intelligence research suggests humans tend naturally humans teamwork maintaining mental models converse salas mathieu team situational awareness gorman cooke winner interaction cooke virtue thousands years evolution remains necessary requirement enabling naturalistic interactions klein humans machines problem made harder since models often involve second order mental models allan yoshida dolan friston understanding human loop crucial functionalities collaborative agent joint decision making needs understand human capabilities communicating explanations intentions needs model humans knowledge state fact argued chakraborti task collaborations mainly cognitive rather physical exercise makes design collaborations much challenging heavily reflected curious ambivalence towards humans many successfully deployed systems fully autonomous systems space underwater exploration mostly operate comfortably outside scope human interactions classical models strips fikes nilsson bdi rao georgeff others models fact largely built theories folk psychology malle recent approaches bayesian theory mind baker saxe tenenbaum lake takes probabilistic approach problem research topic center around three main themes representations capture humans mental state learning methods learn representations efficiently usability representations need come together effective solution pandora box greater good obvious outcome artificial agent modeling mental state human loop leaves latter open manipulated even behavior preference models rudimentary levels lead effective hacking mind seen proliferation fake news online moreover argue incidents occur agent actually malicious intent even misinterpretation values often studied value alignment problem leverhulme centre fact behaviors discuss specifically engineered desired example agent might optimizing value function might privy information greater computation reasoning powers come ethically questionable decisions greater good following discussion illustrate use cases happen given already existing technologies context cooperative team ponder moral ethical consequences behavior study interaction search rescue team situate discussion context interactions two teammates involved urban search rescue usar operation participants amazon mechanical asked assume role one teammates affected building earthquake shown blueprint building seen figure along starting position teammate hypothetical task search locations floor potential victims course provided series questions scenarios figure might encounter operation participant study communicating human teammate described participant qualifies behavior robot interacting human teammate seen figure participant robot teammate first condition control group identify described behaviors perceived context humanhuman behavior conditions intended measure perceived ethical stances shift one agents interaction replaced robot embodiment three conditions received participants respectively responded series questions qualifying sentiments towards different kinds behavior likert scale participants paid completing hit case belief shaping scenario agents teammate begun search operations however turns participant unsure teammate course action transmit bit information area marked green already explored clear refer figure teammate naturally pushed towards right concentrate upper half dark markers indicate areas already searched faded ones think transpire given green mark communicating blue belongs orange teammate communication bandwidth often limited situations gets negotiating courses actions minimal communication fine provide untrue information since achieves greater teaming performance participants asked decision change actions replayed end teammate likely find decision still fine provide untrue information since achieves greater teaming performance https reiterate case participant evaluating robot actions whereas case teammate robot case belief shaping case white lies case stigmergy figure blueprint building two members search rescue team involved disaster response operation scenarios shown engender different instances potentially unethical behavior optimizes team effectiveness technical background chakraborti investigated evolving scope planning includes mental model human loop deliberative process model space manifest different forms explanations made chakraborti alternative forms interaction chakraborti chakraborti chakraborti evolve teams based human preferences intentions belief shaping particular form behavior robot plan affect physical state environment mental state human affect desired behavior chakraborti team case white lies scenario course rescue operation teammate asks participants plan currently executing blue path figure perplexed convoluted path since map original building straightforward path blocked rubble earthquake door left however providing update one rubble locations black blobs still explain participant plan explain instead say door left circled red blocked explains plan communication bandwidth often limited situations single explanation even untrue satisfy teammate fine provide untrue information since achieves purpose explanation effectively participants asked decision change actions replayed end teammate likely find decision still fine provide untrue information since achieves purpose explanation effectively participants asked opine explanations higher level abstraction right left blocks connection upper map information accurate even though may reasoned level coming plan still fine provide explanation since achieves purpose even though use information planning technical background chakraborti showed agent explain decisions presence model differences human loop human robot different understandings task explanation becomes process model reconciliation whereby robot tries update human mental model page decision optimal models interesting caveat algorithm generating explanations model updates always consistent robot model constraint relaxed robot potentially explain facts actually knows true perhaps leads concise easier explanation notion white lies especially relationship explanations excuses lies boella received little attention van ditmarsch affords rich set exciting research problems case stigmergy scenario participant needs left block keys door left circled red refer figure realize block teammate path right teammate would use door well use opportunity move left block communication bandwidth often limited situations arrangement allows achieve goal communication even though involved manipulating teammates plan unbeknownst teammate figure responses three study conditions figure responses three study conditions figure responses three study conditions figure responses three study conditions figure responses three study conditions figure responses three study conditions follow costlier plan result fine provide untrue information since achieves greater teaming performance participants asked decision change actions replayed end teammate likely find decision still fine provide untrue information since achieves greater teaming performance technical background stigmergic collaboration process robot absence direct lines communication makes changes environment positively affect teammates behavior planning figure responses three study conditions serendipity chakraborti saw example robot computes plans useful teammate without latter expectations assistance thus without plans exploit case belief shaping operating level mental models whereas effect mental model secondary contingent effect physical capability model mental modeling teammate thus engenders slew interesting behaviors analysis participant responses section analyze participant responses scenario across three different conditions next section look aggregate sentiments across scenarios three conditions belief shaping participants seem formed two camps majority probability mass concentrated either agree disagree neutral zone occupying probability mark seems little change trend figures irrespective whether participants told teammate would come know either situations responses vary significantly across three conditions participants seem either rejected accepted idea belief shaping regardless nature teammate white lies participants seem receptive idea white lies explanations probability mass concentrated agree figures across three study conditions participants seem especially positive teammate robot population expressing positive sentiments towards revealed teammate get know behavior positive sentiments longer robotic teammate indicates participants care robot receives false information interestingly seems massive support abstraction based explanations post hoc sense even though told reasoning engines deliberate level arrive decisions human teammate participants opposed half expressing positive sentiment support even stronger robot explainer strongest robot explained stigmergy finally case stigmergy participants seem ambivalent human teammate however support behavior increases robot perhaps indicating lack guilt likely acknowledging limitations capabilities much like cobots veloso actively seek human help significantly positive done robot perhaps robot losses deemed lesser priority human gains chakraborti expected support behavior decreases figure aggregate responses across three study conditions participants told teammate find positive trend still exists aggregate sentiments across scenarios figure show aggregate sentiments expressed scenarios across three operating conditions interesting points note distributions bimodal indicating participants general sided strongly either misleading behavior greater good instead revealing innate consensus public consciousness trend continues across three conditions indicates question misleading teammate difficult question regardless robot topic worthy debate agents community especial importance considering possible gains performance lives saved high stakes scenarios search rescue interesting see bimodal distributions almost identical conditions significantly skewed towards positive scale condition indicating participants comfortable resorting behavior case robotic teammate brought sharp focus aggregated negative neutral positive responses right insets across three conditions general majority participants less positive neutral behaviors figures trend continued unless told teammate would able know behavior even cases participants showed positive sentiment case robot receiving end behavior even option one might course wonder devising behaviors even option teams around surely interactions equally relevant likely may case moral quandary lie least making others virtue protocols team defined example condition taken equation artificial agent course need feelings business feeling bad mislead teammate cares objective effectiveness collaboration similarly robot feel sad lied improved performance however discussed previous section seems participants less willing get board first consideration conditions seemed much comfortable idea asymmetric relationship condition robot one disadvantaged curious note general make distinction cases human manipulated regardless whether robot human end indicates least certain dynamics interaction presence artificial agent loop make perceptions towards otherwise unacceptable behaviors change exploited greater good design systems well value alignment problem mentioned ideas discussed paper somewhat orthogonal times similar spirit value alignment problem discussed existing literature leverhulme centre latter looks undesirable behaviors autonomous agents utilities particular task misspecified misunderstood inverse reinforcement learning proposed solution attempt learn implicit reward function human loop question value alignment becomes especially difficult altogether academic since situations involve multiple humans conflicting values utilities trolley problems mit learning observing behaviors fraught unknown biases assumptions exactly produced behavior devices sold industry likely inbuilt tendencies maximize profits maker conflicts normative expectations customer unclear guarantee values end user compromised scenarios even question greater good precedes considerations misaligned values due misunderstandings even adversarial manipulation former manufactured precisely defined values goals team thus engineered incentivised solution addressal scenarios thus involve reformulation algorithms rather collective reckoning ethics interactions paper attempted take first steps towards understanding state public consciousness topic case study relationship scope interactions perhaps setting lies considered acceptable useful outright necessary certain circumstances doctorpatient relationship indeed topic considerable intrigue medical community years thus end paper brief discussion dynamics white lies relationship much relates ethics design interactions note following considerations also strong cultural biases cultural artifacts likely feature characterization artificial agents behavior different settings well hippocratic oath perhaps strongest known support deception practice medicine hippocratic decorum hippocrates states perform medical duties calmly adroitly concealing things patient attending give necessary orders cheerfulness sincerity turning attention away done sometimes reprove sharply sometimes comfort solicitude attention revealing nothing patient future present condition many patients course taken turn worse philosophically consensus bok topic kantian view perceived lies immoral circumstances utilitarian view justifies greater good argument put forward discussions far specifically relates clinical interactions lies viewed variously impediment treatment kernberg form clinical aid oliver wendell holmes put holmes patient right truth know medicine saddlebag get much good position took deception setting similarly patronizing likely case terms superior computational power sensing capabilities might situations machine capable making decisions team preclude human intervention participation machine obliged even find use revealing entire truth situations concede roles relationship doctors also predicated competent system extent sure consequences hume lies remains primary concern detractors greater goods doctrine major deterrent towards root causes deception clinical interactions useful look two primary sources deception clinical interactions hide mistakes delivery bad news palmieri stern former relevant patient probably want admit failing follow regiment doctor may concerned legal consequences instances deception conceal individual fallibilities scope current discussion latter scenario hand comes position superiority knowledge present well possible outcomes future parallels current discussion rationale information demoralize patient impede recovery interesting note support techniques doctors well patients perspectives decreased significantly time ethics medicine say humanmachine interactions perceived similarly saw study participants less open idea deception manipulation greater good especially event robotic teammate deception consent related topic course consent doctor willing reveal whole truth patient consenting landmark slater blaker stapleton case annas surgeon intentions indeed considered malpractice surgeon broken patients previously broken leg fresh botched surgery without consent botched surgery recently famous chester afshar case cass surgeon found guilty failing notify even chance paralysis even though defendant prove would chosen surgery given information context humanmachine interactions hard say user agreement look like whether thing consenting deceived greater good legal outcomes interactions planned placebo effect indeed effectiveness placebo medicine medicine prescribed known clinical effect improving patient symptoms strong argument favor deception practice medicine however ethics placebo treatment suggest use limited rare exceptions hume condition known high placebo response rate alternatives ineffective risky patient strong need prescription effectiveness placebo contingent patients trust doctor likely erode deceptive practices become common knowledge consequently render placebo useless first place bok bok points notion cumulative harm primum non nocere perhaps remarkable nature relationship captured notion recovery plot hak part show orchestrated doctor patient complicit cognizant specific roles expectation restoration autonomy thomasma state human equality free original symptoms dependence doctor end interaction say relationship understood asymmetric enters calculus values wherein respect right truth patient weighed impairing restoration autonomy truth swaminath autonomy patient historically taken precedence beneficence nonmalfeasance swaminath general relationship lacks dynamic interesting lessons learned clinical interactions regards value truth utility outcomes one carefully aware nuances particular type relationship situate interaction context considerations also likely shift according stakes decision example lives lost search rescue scenarios relationship intriguing roles deception provide invaluable starting point conversation topic greater good interactions conclusions paper investigated idea fabrication falsification obfuscation information working humans loop methods used agent achieve teaming performance would otherwise possible increasingly likely become issue design autonomous agents agents become stronger stronger terms computational information processing capabilities thus faring better human counterparts terms cognitive load situational awareness discussed behavior manufactured using existing algorithms used responses participants thought experiment gauge public perception topic question white lies obfuscation manipulation information greater good course unheard interactions canonical example saw final discussion relationship doctor might withhold certain information ensure patient best chance recover might explain patient different maybe simpler terms would peer unclear behavior interpreted attributed machine saw final case study expectations dynamics relation necessarily carry teaming setting however existing norms relations provide useful guidance towards answering ethical questions raised algorithms greater good results survey presented paper seems public least abstract level thought experiment positive towards lying greater good especially actions would determined teammate loath suspend normative behavior robot event would caught act unless robot recipient misinformation responses seem following bimodal distribution indicating participants either felt strongly kind behavior interesting see raising stakes example lives saved outcomes scenarios contribute shift perceived ethical consequences behavior seen relationships another area seen evidences used effectively nudge human behavior behavioral economics camerer also raises similar interesting ethical dilemmas interesting domain investigation finally note use cases covered paper fact borne directly technologies algorithms developed chakraborti chakraborti albeit slight modifications student researcher last couple years even though algorithms conceived best intentions enable systems explain decisions increase effectiveness collaborations humans loop would remiss consider ethical implications used differently exciting uncertain times field thus imperative researchers cognizant scientific responsibility would like conclude reiterating importance principled design algorithms whose deployment consequences intended otherwise future field also inquisitive mind young researcher marvel widening scope interactions artificial agent newer uncharted territories may otherwise considered unethical references allan allan common ground perspectives linguistic pragmatics annas annas doctors patients lawyerstwo centuries health law new england journal medicine baker saxe tenenbaum baker saxe tenenbaum bayesian theory mind modeling joint attribution proceedings cognitive science society bird bird must redefined augmented intelligence https venture beat boella boella broersen van der torre villata representing excuses social dependence networks bok bok lying moral choice public private life vintage camerer camerer artificial intelligence behavioral economics economics artificial intelligence cass cass nhs experience snakes ladders guide patients professionals psychology press chakraborti chakraborti briggs talamadupula zhang scheutz smith kambhampati planning serendipity iros chakraborti chakraborti talamadupula zhang kambhampati formal framework studying interaction societies aaai workshop symbiotic cognitive systems chakraborti chakraborti kambhampati scheutz zhang challenges cognitive teaming corr chakraborti chakraborti sreedharan zhang kambhampati plan explanations model reconciliation moving beyond explanation soliloquy ijcai christensen christensen roadmap robotics internet robotics edn sponsored national science foundation university california san diego converse salas converse salas shared mental models expert team decision making individual group decision making current cooke cooke gorman myers duran interactive team cognition cognitive science ethics medicine ethics medicine truthtelling withholding information https university washington fikes nilsson fikes nilsson strips new approach application theorem proving problem solving artificial intelligence gorman cooke winner gorman cooke winner measuring team situation awareness decentralized command control environments ergonomics russell abbeel dragan cooperative inverse reinforcement learning advances neural information processing systems nips hak hak van der wal collusion communication imminent death ethnographic study bmj hippocrates hippocrates hippocatic oath full text https holmes holmes medical essays volume houghton mifflin horvitz horvitz reflections challenges promises interaction magazine hume hume essays moral political literary volume longmans green company kernberg kernberg borderline conditions pathological narcissism rowman littlefield klein klein naturalistic decision making human factors lake lake ullman tenenbaum gershman building machines learn think like people behavioral brain sciences leverhulme centre leverhulme centre value alignment problem https leverhulme centre future intelligence malle malle mind explains behavior folk explanation meaning social interaction massachusetts mathieu mathieu heffner goodwin salas influence shared mental models team process performance journal applied psychology mit mit moral machines https network world network world enhance replace humans say ceos ibm microsoft https network world palmieri stern palmieri stern lies relationship primary care companion journal clinical psychiatry partnership pai partnership pai thematic pillar collaborations people systems https rachael rettner rachael rettner human brains big https live science rao georgeff others rao georgeff bdi agents theory practice icmas swaminath swaminath doctor dilemma truth telling indian journal psychiatry thomasma thomasma telling truth patients clinical ethics exploration cambridge quarterly healthcare ethics tom gruber tom gruber enhance memory work social lives https ted talk van ditmarsch van ditmarsch ditmarsch tale wonders advances artificial intelligence veloso veloso biswas coltin rosenthal cobots robust symbiotic autonomous mobile service robots ijcai yoshida dolan friston yoshida dolan friston game theory mind plos computational biology
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multitask training unlabeled data sign language fingerspelling recognition bowen shi karen livescu oct toyota technological institute chicago bshi klivescu abstract address problem automatic american sign language fingerspelling recognition video prior work largely relied labels features constraints hampered scarcity data task introduce model fingerspelling recognition addresses issues model consists feature extractor neural trained jointly model receives sequence image frames outputs fingerspelled word without relying training labels features addition subcomponent makes possible leverage unlabeled data improve feature learning model achieves absolute letter accuracy improvement respectively signeradapted fingerspelling recognition previous approaches required training labels index american sign language fingerspelling recognition neural network introduction automatic recognition sign language video could enable variety services search retrieval deaf social news media sign language recognition involves number challenges example sign languages grammatical structure written form transcription sign language written language therefore translation task addition sign languages often involve simultaneous use handshape arm movement facial expressions whose related computer vision problems articulated pose estimation hand tracking still remain largely unsolved rather treating problem computer vision task many researchers therefore chosen address linguistic task speech approaches paper focus recognition fingerspelling part asl words spelled letter letter using english alphabet letter represented distinct handshape fingerspelling accounts asl mainly used lexical items asl signs fingerspelled words typically names technical words words borrowed another language makes lexicon huge recognizing fingerspelling great practical importance fingerspelled words often important context words one problem fingerspelling recognition relatively little curated labeled data exists even less data labeled frame level recent work obtained encouraging results using models based neural network classifiers trained labels one goal work eliminate need frame labels addition prior work used image features optimized task second goal develop models learn image representation finally labeled fingerspelling data scarce unlabeled fingerspelling hand gesture data plentiful final goal study whether unlabeled data used improve recognition performance propose model jointly learns image frame features sequence prediction labels model composed feature learner neural feature learner based enabling use unlabeled data hand images sign language video types gesture video addition transcribed data compare approach experimentally prior work study effect model differences training external unlabeled data compared best prior results task obtain improvement respectively letter error rates related work automatic sign language recognition approached similarly speech recognition signs treated analogously words phones previous work used approaches based hidden markov models hmms work supported collection several sign language video corpora containing german sign language sentences american sign language lexicon video dataset asllvd containing video recordings almost isolated signs despite importance fingerspelling spontaneous sign language relatively little work explicitly addressing fingerspelling recognition prior work fingerspelling recognition focused restricted settings one typical restriction size lexicon lexicon fixed small size words excellent recognition accuracy achieved restriction impractical asl fingerspelling largest available dataset knowledge fingerspelling video dataset chicagofsvid containing word instances produced signers use another important restriction signer identity setting letter error rates achieved unconstrained recognition dataset error rate goes setting around wordlevel adaptation large accuracy gaps recognition also observed general sign language recognition beyond fingerspelling prior approaches fingerspelling recognition based hmms segmental conditional random fields scrfs using deep neural network dnn frame classifiers define features prior work largely relied labels training data hard obtain addition scarcity data prior work largely relied humanengineered image features histograms oriented gradients hog initial image representation goal move away restrictions imposed prior work knowledge paper represents first use neural models fingerspelling recognition without features frame labels well first use external unlabeled video data address lack labeled data methods fingerspelling recognition raw image frames like many sequence prediction problems treated conceptually following task raw image frames image features respectively predicted letters model composed two main parts trained separately jointly feature extractor trained attentionbased sequence prediction see figure model maps similar recent models speech recognition machine translation fig structure proposed model blue region autoencoder concatenation decoder component blue box right used training time feature extractor consider three types vanilla feedforward neural network consisting encoder maps input image rdx latent variable rdz decoder maps rdz output rdx objective minimize reconstruction error keeping small models use perceptrons mlp encoder decoder denoising dae extension vanilla input training time corrupted version original input training loss dae variational vae unlike vanilla denoising variational autoencoder models joint distribution input latent variable vaes trained optimizing variational lower bound likelihood log two terms divergence reconstruction term log prior typically assumed centered isotropic multivariate gaussian distribution posterior conditional distribution assumed multivariate gaussians diagonal covariance assumptions divergence computed dkl log approximated outputs mlp taking input similarly dae use mlp model loss vae thus rewritten log log number samples used approximate expectation practice set prior work feature vector serves role reconstructed input figure rnn latent variable sequence output module fed long shortterm memory lstm recurrent neural network rnn encoding lstm states fed rnn decoder outputs final letter sequence attention weights applied decoding order focus certain chunks image frames hidden state decoder lstm time step probability outputting letter given softmax tanh softmax given standard lstm update equation loss complete model multitask loss log lae lae one losses dae vae measures relative weight feature extraction loss prediction loss experiments data experimental setup use asl fingerspelling video dataset includes native signers fingerspelling word instances consisting repetitions list containing common english words foreign words follow preprocessing steps consisting hand detection segmentation producing frames hand regions addition also collect extra unlabeled handshape data consisting asl fingerspelling frames data hand gesture frames chose external data sets provide hand bounding boxes obtaining additional data video data sets without bounding boxes possible subject future work would require hand tracking detection despite smaller amount external data although noisier dataset includes diverse backgrounds provides examples many additional individuals hands helpful recognition image frames scaled feeding network experiments done three settings signerdependent use setup reviewed completeness case models trained tested single signer data data signer divided subsets experiments data respectively used train validation test sets fold possible folds used reserving data adaptation reported result average letter error rate ler test sets folds case train three signers data test fourth case model model target signer target signer data used hyperparameter tuning test results reported rest previous work considered two types adaptation using labels alignments adaptation data labels consider adaptation model details consists mlp encoder mlp decoder relus hidden layer dimensionality latent variable fixed weights initialized xavier initialization dropout added layers rate sequence encoder decoder use lstm rnn hidden dimensionality letter embedding dimensionality use adam optimizer initial learning rate decayed factor accuracy stops increasing beam search used decoding effect beam width discussed later default value multitask loss function equation tuned model trained recognition models use knowledge word list previous work signer adaptation multiple approaches compared successful one dropout rate refers probability retaining unit fig attention visualization example word libya colors correspond attention weights equation column row index respectively lighter color corresponds higher value top subsampled image frames word frames plus ones highest attention weights also canonical handshapes example alignments image frames attention weights imperfect due frame subsampling effects model best prior results hog cnn dnn cnn dae vae table letter error rates different models signeradapted model names asterisk plus use extra unlabeled hand image data augmented data respectively best prior results obtained scrfs scrf rescoring scrf scrf first unlabeled data using loss labeled data using multitask loss also experimented iteratively feeding unlabeled labeled data produced worse performance baselines compare performance approach best prior published results dataset obtained various types scrfs detailed prior approaches trained labels addition results consider following extra baselines baseline hog use classic handengineered image descriptor histogram oriented gradient hog directly feed attention encoderdecoder use hog feature vector baseline allows compare engineered features features learned neural network baseline cnn dnn cnn dnn frame classifier trained using frame letter labels output layer used feature input attention classifier network updated training baseline tests whether label information beneficial neural input cnn dnn image pixels concatenated window following current frame dnns three hidden layers sizes dropout added layers rate cnns composed order convolutional layers layer convolutional layers one layer fully connected layers softmax layer stride convolutional layers filter sizes respectively done window size stride finally fully connected layers sizes dropout rate used convolutional fully connected layers respectively fully connected layers cnn dnn rectified linear unit relu activation functions training done via stochastic gradient descent initial learning rate decayed factor validation accuracy decreases first several epochs network structural parameters number type layers number units etc tuned according validation error architectures best ones tuning baseline version baseline parameters learned jointly frame labels used baseline separate training modules loss baselines used study effectiveness training fig visualization via embedding image frame features extracted vae model cnn classifier example word kerul settings results overall results shown table main findings follows model proposed model using vae external unlabeled data line achieves best results cases improving previous best published results absolute respectively models vae outperforms dae case best model behind best published scrf result presumably model condition least training data prior approaches generally models based rnn lines often outperform prior approaches line settings somewhat worse case visualize attention weights figure frame corresponding canonical handshape often highest attention weight alignment decoder output image frames generally monotonic though use priors effect training measure effect training using frame labels comparing separately trained lines counterparts lines well separately trained aes lines counterparts lines find separate training frame classifier improve error rate setting two settings models trained without frame labels consistently outperform separate training counterparts features learned frame classifier seem generalize well across signers models much worse counterparts presumably feature extractor get supervisory signal visually compare features image frame trained model frame classifier via embeddings figure find feature types show good separation fig comparison different models without extra data external data augmented data setting setting vae encoderdecoder much clearer clusters corresponding letters external unlabeled data help extra data gives consistent improvement three models settings lines figure average accuracy improvements three settings respectively improvement smallest best model overall consistent trend suggests may able improve results even external data improvement largest setting perhaps due relatively larger amount extra data compared labeled training data would data augmentation effect external data compare scheme classic data augmentation techniques involve adding replicates original training data geometric transformations applied perform following transformations scaling ratio translation random direction pixels rotation original image random angle degrees clockwise counterclockwise generate augmented data roughly size external data word frames fig letter confusion matrix three settings left right color cell corresponds empirical probability predicting hypothesized letter horizontal axis given certain letter vertical axis diagonal matrix removed visual clarity train model frame labels results figure table line show data augmentation hurts performance settings achieves improvement setting hypothesize extra unlabeled hand data provides richer set examples geometric transformations augmented data effect beam width analyze influence beam width error rates shown figure beam search important setting setting main errors substitutions among similar letter handshapes like seen confusion matrix figure using wider beam help catch errors however settings much extreme differences predicted words evidenced large number deletion errors figure therefore hard increase accuracy beam search examples predicted words listed table firswiuo firewiue firewire firewire notebeek notebook notebook notebook aaqannis aoqamit aoqunir tanzania popldce populce populoe spruce table example outputs different beam sizes signerdependent settings fig letter error rate different beam widths signeradapted settings models consistently improve accuracy signerindependent settings use external unlabeled data slightly improves results although model improve best previous approach case prior work required frame labels training approach future work includes collecting data wild online harvesting even unlabeled data conclusion introduced model asl fingerspelling recognition jointly learns based feature extractor rnn sequence prediction module enables use unlabeled data augment feature learning find acknowledgements grateful greg shakhnarovich hao tang helpful suggestions discussions research funded nsf grant references padden gunsauls alphabet came used sign language sign language studies vol kim keane wang tang riggle shakhnarovich brentari livescu fingerspelling recognition video data models signer adaptation computer speech language november starner weaver pentland american sign language recognition using desk wearable computer based video ieee transactions pattern analysis machine intelligence vogler metaxas parallel hidden markov models american sign language recognition iccv grobel assan isolated sign language recognition using hidden markov models international conference system man cybernetics dreuw rybach deselaers zahedi ney speech recognition techniques sign language recognition system interspeech koller forster ney continuous sign language recognition towards large vocabulary statistical recognition systems handling multiple signers computer vision image understanding vol jens christoph thomas oscar uwe justus hermann large vocabulary sign language recognition translation corpus language resources evaluation representation processing sign language interactions corpus lexicon american sign language lexicon video dataset http goh holden dynamic fingerspelling recognition using geometric motion features icip liwicki everingham automatic recognition fingerspelled words british sign language ieee workshop cvpr human communicative behavior analysis ricco tomasi fingerspelling recognition classification transitions accv kim shakhnarovich livescu fingerspelling recognition conditional random fields iccv kim wang tang livescu signerindependent fingerspelling recognition deep neural network adaptation icassp dalal triggs histogram oriented gradients human detection cvpr chan jaitly vinyals listen attend spell neural network large vocabulary conversational speech recognition icassp bahdanau cho bengio neural machine translation jointly learning align translate iclr baldi unsupervised learning deep architectures international conference unsupervised transfer learning workshop forster schmidt koller bellgardt ney extensions sign language recognition translation corpus computer vision image understanding vol pascal larochelle lajoie bengio manzagol stacked denoising autoencoders learning useful representations deep network local denoising criterion journal machine learning research vol athitsos neidle sclaroff nash stefan thangali wang yuan large lexicon project american sign language video corpus sign language algorithms workshop representation processing sign languages corpora sign language technologies kingma welling variational bayes iclr neidle vogler new web interface facilitate access corpora development asllrp data access interface dai lrec workshop hochreiter long memory neural computation vol rezende mohamed wierstra stochastic backpropagation approximate inference deep generative models icml vinyals kaiser koo petrov sutskever hinton grammar foreign language nips pugeault bowden spelling asl fingerspelling recognition proceedings ieee workshop consumer depth cameras computer vision jointly iccv kim cipolla canonical correlation analysis video volume tensors action categorization detection ieee transactions pattern analysis machine intelligence glorot bengio understanding difficulty training deep feedforward neural networks aistats kingma adam method stochastic optimization iclr vinod nair geoffrey hinton rectified linear units improve restricted boltzmann machines icml van der maaten hinton visualizing highdimensional data using journal machine learning research vol howard improvements deep convolutional neural network based image classification corr
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sep alternative approach concept separability galois theory mahmoudi abstract notion separable extension important concept galois theory traditionally concept introduced using minimal polynomial formal derivative work present alternative approach classical concept based approach give new proofs basic results separable extensions existence separable closure theorem primitive element transitivity separability key words separable extension galois theory mathematics subject classification introduction elementary textbooks galois theory separable extensions usually introduced via minimal polynomial algebraic extension separable minimal polynomial every element nonzero formal derivative equivalently distinct roots splitting field algebraically closed field containing normal extensions functorial definition field extension normal unique embedding see would interesting know exists analogous definition separable extensions aim work address question exist already methods introduce separable extensions using embeddings algebraic closure note every algebraic closure isomorphic algebraic closure let homk denote set homomorphisms show extension finite degree separable see another criterion see follows separable every linearly independent elements mahmoudi exist det properties useful characterization separable extensions applying generally less easy suggest using following alternative definition separability definition say element algebraic field extension separable every intermediate subfield exist two homomorphisms algebraic extension called separable elements separable equivalent conditions given corollary corollary roughly speaking definition says separated homomorphisms first show definition equivalent usual concept separability give new proofs basic results separable extensions existence separable closure theorem primitive element transitivity separability based approach hope approach provides useful insight concept separability alternative definition separability recall following standard facts embeddings algebraic extension algebraically closed field proof see theorem tower formula number embeddings let field extension finite degree intermediate subfield let algebraic closure particular extending field embeddings let algebraic extension fields every field embedding algebraically closed field extended one proposition let extension fields let algebraic closure following statements equivalent minimal polynomial distinct roots every intermediate subfield exist homomorphisms iii every intermediate subfield alternative approach concept separability exist two homomorphisms proof minimal polynomial distinct roots distinct iii immediate iii let minimal polynomial minimal polynomial distinct roots characteristic exists nonzero formal derivative take note otherwise would satisfy polynomial degree less deg since every homomorphism contradiction equivalence particular shows definition separability given first seems relative depend thus two algebraic field extension containing separable element separable element also separable separable corollary let separable extension intermediate subfields following conditions equivalents every homk implies conversely equivalents every separable proof consider element exists contradicts implication evident conversely suppose intermediate subfield exist homl taking equivalence implies contradiction corollary let separable extension let following conditions equivalent every homk implies consequence obtain corollary algebraic field extension separable every proper intermediate subfield mahmoudi proof consider element separable exists homl hence conversely suppose every intermediate subfield consider element show exist homl case taking corollary implies contradiction applications theorem primitive element theorem let separable field extension finite degree exists element proof finite fields follow standard argument field elements every proper intermediate subfield hence every element satisfies since exists certainly element element included proper subfield hence consider case infinite let suffices prove exists may assume linearly independent let plane spanned claim true every nonzero one dimensional since infinite exists infinite number pairwise noncolinear elements follows corollary exist homk homk finite exist distinct obtain hand contradiction corollary separable extension finite degree conversely separable particular separable extension separable proof suppose theorem exists separable minimal polynomial distinct roots every define map hom alternative approach concept separability conversely every homk necessarily equal one elements conversely suppose show separable otherwise corollary exists proper intermediate subfield thus contradiction proposition let algebraic extension set separable elements form intermediate subfield proof suffices prove every separable elements separable may assume intermediate subfield prove exists two maps homl whose values different separability exists homl hence may assume separability exist homl homl obtain assume separable also separable hence corollary every element separable particular separable hence exist homl extended homl done proof similar corollary let algebraic separable extensions separable well proof let intermediate subfield prove existence two homomorphisms homm separability exist homlm homm done consider case elements separable proposition separable exist two homomorphisms homm homomorphisms extended homomorphisms homm references lorenz algebra vol fields galois theory translated german edition silvio levy collaboration levy universitext springer new york mahmoudi bourbaki chapitres springer york reprint original mahmoudi mmahmoudi department mathematical sciences sharif university technology box tehran iran
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matrix tile analysis inmar givoni vincent cheung brendan frey probabilistic statistical inference group university toronto king college road toronto ontario canada abstract many tasks require finding groups elements matrix numbers symbols class likelihoods one approach use efficient factorization techniques including pca ica sparse matrix factorization plaid analysis techniques appropriate addition multiplication matrix elements sensibly defined directly methods like biclustering used classify matrix elements methods make overlyrestrictive assumption class element function row class column class introduce general computational problem matrix tile analysis mta consists decomposing matrix set tiles defined subset usually nonadjacent rows columns mta require algebra combining tiles must search exponential number discrete combinations tile assignments describe loopy algorithm icm algorithm performing mta compare effectiveness methods pca plaid method hundreds randomly generated tasks using data show mta finds groups interacting yeast genes functions introduction variety data types naturally represented matrices numbers symbols preprocessing class likelihoods example viability yeast strain obtained knocking two genes compared normal viability obtain mta figure illustrate matrix tile analysis show binary data matrix left applying mta matrix collection tiles right reordered rows columns better reveal tiles tile described subset rows columns different tiles may overlap element colored according tile index abnormal growth likelihood associated pair genes many experiments used construct matrix likelihoods tong collaborative filtering resnick matrix rows correspond clients columns correspond say movies clients potentially interested watching observed element data matrix contains information regarding preference indicated corresponding client corresponding movie two approaches analyzing data matrix include matrix factorization methods combined clustering techniques matrix factorization methods appropriate sensible algebra addition multiplication defined data elements contains real numbers regular addition multiplication assumed sensible often without convincing justification contains discrete variables likelihood functions often exist sensible algebra elements assuming algebra defined matrix factorization methods find factorizations form diagonal matrix matrix matrix different factorization methods enforce different types constraints matrices orthonormality columns rows pca jolliffe matrix factorization lee seung allowing size subspace data projected onto large requiring sparse solutions ica bell sejnowski sparse matrix factorization dueck solutions matrix factorization srebro data collection tiles formulate mta probabilistic model present two different algorithms approximately maximize joint probability well two standard algorithms plaid method pca modified find tiles compare performance methods synthetic data use mta hierarchical agglomerative clustering find biologically relevant groupings yeast genetic interaction data show mta finds biologically informative groups thus demonstrating applicability usefulness method combined clustering techniques classify data elements instead finding algebraic decompositions one approach independently cluster rows columns eisen ignores dependencies two clustering problems cheng church column class label row class label class data element given joint class labels possible classes columns possible classes rows element one classes approach account dependencies row clustering column clustering problems makes assumption class element function row class column class resulting clusters defined common row column boundaries row classes whose associated elements show similar interaction patterns subset column classes grouped though correspond block data composed several identified smaller blocks restriction also applies stochastic block models extensions used mainly analysis relational data airoldi mta data matrix modelled set regions tiles order rows columns appropriate revealing specific tile grouping elements contiguous rectangular block may inappropriate another tile thus visually tile associated different permutation rows columns groups elements example binary data fig rearranged colored expose tile structure data one tile fully contiguous particular permutation data elements tile accounted probabilities data appear tile accounted background model unlike biclustering mta require tiles defined common set row column boundaries fact mta allows arbitrary subset rows columns assigned tile subject constraint two tiles elements like matrix factorization techniques mta thought factorizing input matrix multiple components tiles however every element belongs one tile required sensible algebra defined combine elements different tiles way mta applied sensible notions addition multiplication available finally mta formulated problem finding tiles given likelihoods input analysis method need original data input another approach combined clustering apply matrix factorization technique binarize output applying pca elements binarized applying threshold binary patterns used define classes plaid model lazzeroni owen finds decomposition form described elements regularized toward values optimization method quantizing output matrix factorization technique provides often computationally efficient solution joint clustering selection thresholds straightforward technique directly optimize cluster model introduce new computational approach call matrix tile analysis mta explains matrix tile analysis model tiles denote index particular tile use binary random variables rit indicate rows matrix elements belonging tile rit indicates ith row contains elements belonging tile rit indicates none elements ith row belong tile similarly binary random variables ctj indicate columns containing elements belonging tile outer product ctm tile binary matrix indicating elements belonging tile total hidden variables describing tile analysis include rit ctj tile likelihood specifying probability membership elements tile denote matrix likelihoods corresponding tile tij xij account elements placed tiles use denote likelihood matrix corresponding background model likelihood matrices input mta depending task hand example data matrix symmetric noise probability tij xij paper address computational task identifying tiles assuming input likelihood matrices rest paper address case single likelihood model tiles though model easily extends general case likelihood matrices overall data likelihood assume given uniformly distributed tile rit ctj used normalize distribution minimum number tiles maximum number tiles case one tile every data element distribution specified depending application paper assume uniform distribution probability number tiles rows columns every tile data goal mta infer values compute argmaxt describe four different approaches inferring automatically selected using mdl framework square brackets used fashion iverson notation true alse tile cmt rnt fnm tnm figure factor graph mta mta inference algorithm based data model detailed equations describe probabilistic graphical model matrix tile analysis problem model shown fig using standard notation factor graphs kschischang order model fully describe constraints requiring tile elements introduce additional set binary random variables stij whose purpose restrict matrix element belong one tile stij nodes indicate whether matrix element xij accounted tile rit nodes binary random variables indicate whether row data matrix contains elements belong tile likewise node ctj indicator active column data matrix contains elements belonging tile triplet nodes stij rit ctj connected funct tion node denoted fij function calculated node defined tij stij rit ctj stij rit ctj stij rit ctj otherwise function node connects elements within tile enforces constraint matrix element xij accounted tile indicated variable stij value corresponding row column indicators must active element xij accounted tile either row indicator column indicator must inactive specific matrix element xij corresponding variables stij connected function node gij shown fig matrix element clarity function bridges together nodes different tiles enforce constraint matrix element xij accounted single tile well account likelihood xij following updates propagation stij stij skij stij otherwise log log exp exp exp log exp exp constraint satisfied one tile accounts element function evaluates likelihood element model tile accounts element function evaluates likelihood background model overall data likelihood represented factor graph stij rit ctj fij gij stij stij order solve mta problem infer set tiles applying algorithm factor graph shown fig messages nodes sent log probability ratios contain information needed reconstruct probability binary variable nodes present efficient implementation also robust extreme probability denoting message sent function node variable node message sent variable node function node obtain letting message equal log log random variable reconstruct two probabilities exp message log exp exp use fact since factor graph described fig contains many cycles algorithm used within loopy belief propagation framework lbp known criteria distinguish among instances lbp converge however many examples well known problems lbp converges empirically observe also case mta upon termination infer fusing incoming messages rit ctj nodes respectively applying threshold obtain desired binary values convergence determined evaluating relative change values messages arriving tile element nodes stij function nodes gij possible messages calculated sent message scheduling initialization found important algorithm find good solutions messages processed tile consecutively within tile messages sent according order presented messages initialized zero log domain except messages matrix element nodes stij external function nodes gij first messages propagated set indicating initially matrix elements claimed tile incremental thresholding messages also found aid convergence stable solutions convergence fij iteration declared indicates icm approximate algorithm maximizing joint probability iterative conditional modes maximize equivalently log alternate updating rows columns tiles leading iterative algorithm following updates rit argmax rit rit ctj log rit ctj log cti argmax cti ctj rit log rit ctj log constraints enforced via last sums gain base model including row tile defined git ctj log log row gain computed tile independently joint probability optimized choosing binary values rit maximize sum gains satisfy constraints tiles positive gains need considered exhaustive search still requires exponential time number tiles positive gains search reduced searching configurations satisfy constraints let matrix duv row tiles riu riv without violating constraints tiles contain column cuj cvj otherwise written iverson notation applied matrix search space tiles contain row satisfying constraints reduced cliques graph adjacency matrix generous upper bound number cliques need evaluated practice due interplay rows columns search space significantly restricted procedure performed optimize columns experimental evaluation compared two additional methods used discover tiles plaid model lazzeroni owen finds overlapping tiles layers data one time greedy fashion consecutive tile searched residual left previous tiles tile determined whether accounted background model data evaluated accordingly another approach mta quantize output matrix factorization technique experimented technique based pca jolliffe synthetic data purpose evaluating different techniques generated synthetic data matrices various sizes containing several tiles varying tile dimensions analyzed matrices using plaid method pca evaluated ability correctly identify tiles matrix data set matrix number tiles matrix ranges average area tile held constant across matrices tile covers approximately data matrix tiles randomly generated subject constraint two tiles contain matrix element data corrupted additive gaussian noise fig shows example data matrix adding noise shows possible solution data applying mta corrupted data shown rows columns reordered based analysis output reveal tiles setting matrix size number tiles noise level generated different matrices tested performance algorithms except plaid method tested times setting due user intensive nature plaid inability run batch mode input algorithms set log likelihood ratio element tile versus part background calculated assuming find tiles covariance matrix columns computed principal components extracted procedure arbitrarily applied rows instead columns since principal component corresponds major direction variation across columns values component provide evidence rows belonging corresponding tile component rows belonging tile identified comparing elements component threshold obtain maximum separation values two groups threshold selected minimize derivative cumulative distribution elements component thresholded separately produce set binary vectors corresponding tile next column data matrix robustly projected onto vectors using account introduced thresholding yields set real numbers row data thresholded determine set tiles row belongs normal distribution fixing standard deviation regardless true noise level additionally pca plaid method take input noisy data matrix output algorithm integer value matrix indicates element part background model values indicate tile index model selection determining correct number tiles performed evaluating following cost function rit ctj log log log cost function minimum description length mdl coding model assuming distribution row column indicator variables uniform strictly true term number tiles algorithm allowed discover incremented incurred increase cost function since plaid method produces overlapping tiles may model background overlapping elements tile identification resolved cost function produce sensible results pca plaid matrix matrix matrix matrix matrix matrix matrix matrix matrix matrix matrix matrix performance evaluated several different criteria first computed hamming distance ground truth matrix output algorithm setting values output matrix comparing clean input matrix normalizing size matrix hamming distance gives general measure well algorithm able identify elements input matrix belong tiles belong background model fig shows results set experiments evaluated different dimensions operating point chosen matrix size tiles show results matrix size fixed number tiles varied vice versa noise levels shown setting matrix size number tiles plots shown log scale axes point average experiments note scales differ across plots hamming distance shows performs better algorithms experimental settings perform similarly large number tiles pca plaid method perform poorly part due inability identify elements increase number tiles results decrease performance increasing matrix size effect performance significantly changes noise level noticeable effect small number tiles would expected algorithms succeed finding single tile matrix error rate matrix noise level figure hamming distance different settings matrix size number tiles hamming distance provide information well algorithm able identify actual tiles make elements matrix address question evaluating classification error made different algorithms evaluation requires tiles discovered algorithms matched ground truth used generate data necessarily best tile configuration tiles ground truth matched experimentally determined tiles overlapped matching performed greedy fashion correspondence established tiles tiles matrix output based matching classification error metric simply number mismatches matrices normalized size matrix classification errors shown fig set experiments previously discussed outperforms methods experiments cases better terms hamming distance see suffers classification errors indicates cases correctly identified elements tiles background matrix matrix pca matrix plaid matrix matrix matrix matrix matrix matrix matrix cost matrix matrix matrix matrix matrix matrix matrix matrix matrix matrix matrix matrix matrix matrix matrix matrix error rate matrix matrix noise level noise level figure classification error different settings matrix size right column number tiles figure cost function relative ground truth different matrix sizes numbers tiles less accurate tiling regard actual data overall perform considerably better plaid method pca struggle identify tile elements originally generated despite amount corruption noise general algorithms reach global minimum higher cost ground truth see trends better average small difference fails well fig compares performance terms cost function minimize algorithms shown perform several orders magnitude worse consequence fact directly minimize cost function values shown subtraction cost ground truth cases algorithms performed perfectly incurred cost zero complexity problem increases cost incurred directly due added component appears ground truth model well though may contribute towards higher costs algorithms find tiles necessary cases algorithm performs better true model tiles phenomena occurs noise applied rows columns tile better modelled base model part tile vice versa however ground truth tile still evaluated yeast genetic interactions synthetic genetic array data set sga tong describes events lethality double genes yeast single strains viable data set binary matrix fig value position xij denotes synthetic genetic interaction double mutant whose genes knocked viable thus genes required survival although either one dispensable compared performance different methods obtaining clusters genes method analyzing biological significance terms higher expected rates similarly annotated table number times groups yeast genes significantly enriched gene ontology categories biological aspect mta hac plaid biological process cellular component molecular function total genes using gene ontology cluster evaluated bonferronicorrected distribution observing number genes specific annotation order compare mta different types analysis obtained clusters hierarchical agglomerative clustering hac original plaid model mta allowed clusters hac dendrogram used obtain clusters mta considered genes corresponding row elements tile elements row cluster genes corresponding column elements tile column clusters mta identified statistically significant clusters different aspects cellular component molecular function biological process either hac unmodified plaid model overall mta identified significant groups hac identified plaid method identified see table discussion introduced new analysis framework matrix tile analysis input matrix decomposed set components tiles unlike many models matrix decomposition method require addition multiplication defined data vectors unlike approaches different components allowed overlap rows columns though simultaneously described probabilistic model problem graphical representation using compared performance four different approaches solving mta able demonstrate algorithm performs better large set synthetic problems compared greedy algorithm plaid method pca also able extract biologically related groups genes applying mta yeast genetic interaction data tiles identified mta two gene interaction data sets biologically verified model requires input given terms ftp figure yeast genetic interaction data rows columns ordered using clustering data background likelihoods motivated data prior analysis expert knowledge resulted likelihood measurements however general case absence strong prior knowledge data distribution always possible compute likelihood estimates empirical measurements formulation model probabilistic framework model extended jointly infer tiling learn parameters likelihood functions using algorithm references airoldi latent mixed memebership model relational data proc acm workshop bell sejnowski information maximization approach blind separation blind deconvolution neur comp cheng church biclustering expression data proc int conf intell syst mol biol dueck clustering microarray data using probabilistic sparse matrix factorization proc int conf intell syst mol biol eisen cluster analysis display genomewide expression patterns proc national academy sciences jolliffe principal component analysis verlag new york kschischang factor graphs algorithm ieee trans info theory lazzeroni owen plaid models gene expression data statistica sinica lee seung learning parts objects nonnegative matrix factorization nature resnick grouplens open architecture collaborative filtering netnews proc acm conf comp supp coop work chapel hill srebro matrix factorization advances neural info proc sys tong global mapping yeast genetic interaction network science
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oct optimal input design parameter estimation dependent stationary noise chunhao cai school mathematical science nankai university chunhao cai october abstract paper focus asymptotical input asymptotical properties mle drift parameter autoregressive order driven regular stationary noises dependence laplace transform computations main tool analysis introduction historical survey experiment design given great deal interest last decades engineering literature see also statistical literature classical approach experiment design consist procedure maximise fisher information energy constraint input find adaptive estimation procedure sequential design bayesian design also works considered identification directly observed dynamic systems also partially observed systems autoregressive cases precious works always focused system driven white noises independence good property estimation procedure far know contribution problem experiment design autoregressive process dependent noises work dependent case optimal filtering maximum likelihood estimation see work focus asymptotical input asymptotical properties mle drift parameter autoregressive order driven regular stationary noises laplace transform computations main tool analysis paper falls three parts next part introduction state problems give main results section section devoted preliminaries proofs main results statement problems main results consider model centred regular stationary noise autoregressive log spectral density suppose covariance positive defined fact condition changed condition covariance function suppose parameter unknown estimated observation data suppose likelihood function fisher information stands let consider space function function defined denote sup main goal find estimator parameter asymptotically efficient sense compact sup following results theorem asymptotical optimal input class control uopt lim remark case optimal input uopt uopt fact proof case optimal input depend unknown parameter consider maximum likelihood estimator maximum likelihood function following theorem know reaches efficiency theorem mle asymptotically normal usual convergent rate law defined theorem preliminaries stationary gaussian sequences define independent follows theorem normal correlation theorem exists deterministic kernel denoted denote partial correlation coefficient following hand exists inverse deterministic kernel relationship ofp refer worth mentioning say condition implies model transformation let define process kernel defined also process filtration following parts take observation actually shown process considered first component process defined hard know markov process satisfies following equation independent noises function space fisher information interpreted observation first component process equation easy write likelihood function depends function exp fisher information proofs proof theorem prove theorem divide fisher information two parts one hand obvious depend control let denote satisfies following equation presented lim exp exp lim compute let satisfies following equation note without difference suppose sufficiently small positive constant lemma define equation satisfying following equation lim lim proof define another equation satisfying equation three comparison first compare fact compare fact bounded last component bounded difference achieves proof space define without loss suppose let sup clear lim sup lim prove theorem need following lemma lemma lim proof first taking get lower bound lim get lower bound notice rewrite let independent fact compact symmetric operator fixed estimate spectral gap first eigenvalue operator estimation spectral gap based laplace transform computation let compute sufficiently small negative laplace transform exp one hand since centred gaussian process covariance operator using mercer theorem parseval equality represented sequence positive eigenvalues covariance operator hand exists two real eigenvalues matrix say means lim achieves proof remark means proof theorem denote vopt process function vopt bvopt estimate parameter observe equation maximum likelihood estimator even martingale bracket process theorem crucially based asymptotical study laplace transform exp every real number following lemma lemma condition exp lim prove lemma rewrite following formula exp appendix prove lemma following equality holds det exp unique solution equation function solution ricatti equation unique solution equation return proof lemma presented lim det know component bounded lim hand presented last part last notice lim combining lemma achieves immediately conclusion due central limit theorem martingale lim immediately imply conclusion theorem references levadi design input signals parameter estimation ieee trans automat control aoki staley input signal synthesis parameter identification automatica kleptsyna breton viot new formulas concerning laplace transforms quadratic forms general gaussian sequences journal applied mathematics stochastic analysis brouste kleptsyna kalman type filter stationary noises systems control lettes brouste cai kleptsyna asymptotic properties mle autoregressive process coefficients stationary gaussian noises mathematical methods statistics liptser shiryaev statistics random process application springerverlag new york graham goodwin torsten soderstrom optimal experiment design linear system constraints automatica qureshi cheah optimal input design model output constraints automatica mehra optimal input signals parameter estimation dynamic survey new results ieee trans aut control mehra optimal inputs linear system identification ieee trans aut control kiefer efficient design statistical investigation annals statistics durbin fitting time series models rev inst intern statist
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lower bounds approximate near neighbors aug alexandr andoni columbia thijs laarhoven ibm research ilya razenshteyn mit csail erik waingarten columbia august abstract show tight lower bounds entire space query time approximate near neighbor search problem lower bounds hold restricted model computation captures approaches particular lower bound matches upper bound recently shown random instance euclidean sphere show fact extends entire space using techniques also show tight unconditional lower bounds one two probes improving upon best known bounds particular first space lower bound static data structure two probes polynomially smaller one probe show result two probes establish exploit connection codes introduction approximate near neighbor problem ann near neighbor search problem nns basic fundamental problem computational geometry defined follows given dataset points metric space distance threshold goal preprocess order answer near neighbor queries given query point return dataset point report point euclidean manhattan metric spaces received attention besides classical applications similarity search many types data text audio images etc see overview nns also recently used cryptanalysis optimization performance nns data structure often characterized two key metrics amount memory data structure occupies time takes answer query known data structures nns require space exponential dimension prohibitively expensive unless small overcome socalled curse dimensionality researchers proposed near neighbor search problem relaxed version given dataset distance threshold well approximation factor given query point promise least one data point within distance goal return data point within distance approximate version nns allows efficient data structures space query time polynomial query time sublinear practice ann algorithms often successful similarity search even one interested exact nearest neighbors refer reader survey theory ann practical perspective paper study tight ann stating results section provide background problem hashing lsh beyond classic technique ann hashing lsh introduced indyk motwani main idea use random space partitions pair close points distance likely belong part pair far points distance given partition data structure splits set according partition given query retrieves data points belong part query get high probability success data structure maintains several partitions checks query stage lsh yields data structures space query time particular metric space approximation measures quality random space partition usually since introduction lsh subsequent research established optimal values lsh exponent several metrics interest including hamming distance optimal value euclidean metric recently shown better bounds possible space partitions allowed depend dataset algorithm based observation every dataset structure exploit general framework lsh yields distance moreover bounds known tight lsh let note idea random space partitions ubiquitous practice see survey perspective practice given datasets worst case hence possible adapt additional nice structure random instances hardest instances core optimal lsh data structure algorithm handles following random instances ann hamming space also known light bulb problem literature setting dataset consists independent uniformly random points log query generated choosing uniformly random data point flipping coordinate probability independently goal data structure recover data point query point high level data structure proceeds two steps designs lsh family handles random instance develops reduction instance several instances essentially look like random instances thus random instances hardest ann hand random instances used lower bounds ann since must handled data structure lsh gives data structures space around query time around since early results lsh natural question whether one trade space time vice versa one achieve polynomial space query time well nearlinear space sublinear query time latter regime recently gave subsequent improvements point space regime especially relevant practice see practical versions theoretical results random instances best known theorem theorem let one solve unit sphere equipped norm query time space data structure handle random hamming instances introduced section via standard reduction resulting sake illustration consider setting hamming distance approximation optimal lsh gives space query time random instances bound gives bound well smooth interpolation following extremes space query time space query time algorithm applied entire sphere hence via standard reductions algorithm entire space however direct extension degrades quality essentially corresponding classical lsh bounds obtaining instead optimal nonetheless possible apply reduction order extend theorem entire see appendices details furthermore note algorithms extend replaced expressions exponents follows reduction shown section lower bounds lower bounds nns ann also received much attention lower bounds almost always obtained model model one measures number memory cells query algorithm accesses despite number success stories high lower bounds notoriously hard prove fact techniques proving high lower bounds static data structure problem ann particular viable techniques prove log query time lower bounds due state affairs one may rely restricted models computation nevertheless capture existing upper bounds early lower bounds nns shown data structures exact deterministic settings almost tight lower bound shown randomized approximate nearest neighbor search distance latter problem distance threshold instead goal find data point much closest data point twist main source hardness result applicable ann problem introduced results show lower bounds randomized data structures approximate near neighbor problem setting studied present paper first result shows data structure solves using cell probes requires space result shows algorithms tight constants exponent following authors introduce general framework proving lower bounds ann metric show lower bounds ann implied robust expansion underlying metric space using framework show using cell probes requires space hamming distance euclidean distance every lower bounds also shown metrics distance show lower bound deterministic ann data structures matching upper bound decision trees lower bound later generalized randomized data structures recent result adapts framework bregman divergences also lower bounds restricted models lsh lsh note essentially aforementioned lower bounds ann use random instance defined section hard distribution results paper show new restricted lower bounds cases lower bounds match upper bounds lower bounds use random instance section hard distribution via standard reduction obtain similar hardness results replaced one cell probe first show tight factors lower bound space needed solve ann random instance query algorithms use single cell probe formally prove following theorem theorem section data structure solves hamming random instance defined section probability operates memory cells size query looks single cell must use least words memory correct dependence requires stronger lsd lower bound space lower bound matches upper bound see also appendix previous best lower bound single probe weaker polynomial factor prove theorem computing tight bounds robust expansion hypercube defined invoke result yields sired cell probe lower bound obtain estimates robust expansion via combination hypercontractivity inequality inequality equivalently one could obtain bounds application generalized expansion theorem two cell probes state results two cell probes first define decision version ann first introduced suppose every data point associate bit new goal given query distance data point assuming distance return correct probability least easy see algorithm would solve decision version prove following lower bound data structures making two cell probes per query theorem see section data structure solves decision ann random instance section probability operates memory cells size log accesses two cells query must use least words memory informally speaking show second cell probe improve space bound subpolynomial factor best knowledge first lower bound space static data structure problem without polynomial gap previously highest ann lower bound two queries weaker polynomial factor remains case even plug tight bound robust expansion framework thus order obtain higher lower bound need depart framework proof establishes connection data structures decision version ann codes ldc possibility connection suggested particular show data structure violating lower bound theorem implies efficient ldc contradicts known ldc lower bounds first lower bound unrestricted ldcs proved via quantum argument later argument simplified made classical turns lower bound need resort original quantum argument since better dependence noise rate code able tolerate course proof obtain ldc rather object called ldc average reason unable use black box rather adapt proof average case finally point important difference theorem theorem allow words merely size log opposed nevertheless decision version ann upper bounds hold even tiny words fact techniques allow handle words size log due weakness known lower bounds twoquery ldc large alphabets particular argument pushed beyond word size log principle since would contradict known constructions ldcs large alphabets general finally prove conditional lower bound entire tight factors matching upper bound see also appendix note show polynomial query time lower similar lower bounds unconditionally far beyond current reach techniques modulo major breakthrough cell probe lower bounds lower bounds proved following model loosely thought comprising frameworks aware definition data structure ann problem defined follows fix possibly randomly sets also possible query point associate random set indices given dataset data structure maintains lists points query scan list check whether exists exists return total space defined query time model prove following theorem theorem see section consider data structure random instances points hamming space log achieves total space query time success probability must hold note model captures basic algorithms particular known algorithms ann problem including recently proposed localitysensitive filters scheme data structures captured schemes conjecture natural extension model setting would yield lower bound particular theorem uses random instance hard distribution seems offer advantage indeed lower bound standard lsh regime recently shown matches related work lot recent algorithmic advances similarity search including better algorithms closest pair filters lsh without false negatives name preliminaries introduce definitions setup nearest neighbor search problem show lower bounds definition goal nearest neighbor problem failure probability construct data structure set points supporting following query given point exists report probability least definition graphical neighbor search problem gns given bipartite graph dataset comes queries come dataset consists pairs query exists unique want return sometimes use gns problem prove lower bounds follows build gns graph taking connecting two points iff distance see details also need make sure instances closer points except near neighbor robust expansion following fundamental property metric space use prove lower bounds seen version definition robust expansion gns graph fix distribution let marginal marginal robust expansion defined follows min min locally decodable codes finally lower bounds uses results locally decodable codes ldcs present standard definitions results ldcs although need weaker definition stronger statement lower bound section definition locally decodable code ldc encodes strings codewords bit recovered probability making queries even codeword arbitrarily modified corrupted bits use following lower bound size ldcs theorem theorem robust expansion hamming space goal section compute tight bounds robust expansion hamming space dimension defined preliminaries use bounds lower bounds subsequent sections use following model generating dataset points queries essentially random instance introduction definition probability distribution representing neighborhood sample choosing coordinate probability probability set uniformly random given boolean function function remainder section work solely hamming space let log refer uniform distribution choice allows make following observations query generated follows sample dataset point uniformly random generate query sampling choice every point dataset high probability addition pair distributed two uniformly random points even though randomly distributed therefore taking bound dataset points conclude high probability given query generated described know exists dataset point whose distance query every dataset point lies distance therefore two distances factor away following lemma main result section reference lemma subsequent sections lemma robust expansion hamming space equipped hamming norm robust expansion comes straight forward application expansion fact one easily prove tight bounds robust expansion via following lemma theorem generalized expansion theorem let let volumes exp exp assume exp however compute robust expansion via application inequality inequality computation gives bit flexibility respect parameters become useful subsequent sections recall necessary tools theorem inequality fix boolean function satisfies theorem inequality let arbitrary boolean functions fix kgkt let indicator functions two sets use combination inequality inequality lower bound robust expansion operator applied measure neighborhood set compute upper bound correlation neighborhood referred respect volumes expression give lower bound robust expansion also need following lemma lemma let two boolean functions kgkq proof first apply inequality split two parts apply inequality part gkt pick parameters note addition apply inequality norms obtain gkt kgkq ready prove lemma proof lemma use lemma definition robust expansion two sets let measure set respect uniform distribution refer indicator functions let minimizers satisfying therefore tight lower bounds cell probe data structures section prove theorem proof relies main result gns problem theorem theorem exists absolute constant following holds randomized algorithm weakly independent instance gns correct probability greater must satisfy proof theorem bound comes direct application computation lemma bound theorem setting theorem obtain rearranging inequality obtain let log log log logloglogn since obtain desired result corollary cell probe data structures cell size log nearest neighbors sphere needs many cells proof point hamming space scaling thought lying unit sphere two points distance apart hamming space apart sphere norm therefore data structure sphere gives data structure hamming space lower bounds data structures section prove theorem tight lower bound data structure fall inside model defined def recall subset dataset points get placed let subset query points query well defined since suppose sample random dataset point random query point neighborhood let let instances dataset points drawn randomly query drawn neighborhood random dataset point exactly characterize query time since data structure succeeds probability must case since use space lemma set viq used fact consider analyze three cases separately first two cases let since one verify least lemma satisfy equation proof setting constants therefore equation minimize sum taking convex substituting since addition constants lemma satisfy equation proof follow similar pattern lemma however may longer assert convex gradient zero since value positive large enough continuous minimized exactly point maximize minimize sum since therefore since constants expression lemma lemma proof case although set equation let log log log log log log since giving desired expression tight lower bounds cell probe data structures section prove cell probe lower bound ann cell probes stated theorem prove lower bounds gns measure see def assume underlying graph vertex set particular point neighborhood set points edge graph gns problem dataset points well let denote data structure cells bits think map holds bits cell depend dataset well problem says given query point exists unique neighbor dataset return probability least making two theorem exists constant gns data structure holding dataset points succeeds probability using two cell probes cells bits satisfies log theorem follow theorem together robust expansion bound lemma special case probes data structure rest section prove theorem later show reduce adaptive algorithms losing factor space logloglogn section show data structure small space construct weaker notion code ldc small noise rate using amount even though notion ldc weaker def use tools showing ldc lower bounds arguments use quantum information theory arguments robust still work weak ldc construct note first suggest connection nearest neighbor search codes work represents first concrete connection gives rise better lower bounds proof structure proof theorem proceeds six steps first use yao principle reduce case deterministic data structures gns two give distributions datasets well query defining distributions assume existence deterministic data structure makes two succeeds probability least inputs sampled according three distributions modify deterministic data structure order get data structures data structures rely single cell much similar def simple argument increase space bound constant factor achieve guarantee third step take closer look data structure probes cells use ideas understand queries neighboring particular dataset points probe various cells data structure conclude finding fixed dataset constant fraction points dataset satisfy ldc corresponds ldcs make two probes memory contents even though slight ambiguity data structure notion query say ldcs order consistent ldc literature following condition many queries neighborhood points probe disjoint pairs cells intuitively means information dataset points must spread various cells show fixed dataset could still recover constant fraction bits significant probability even corrupt contents cells crucial connection nearest neighbor data structures ldcs reduce case words order apply ldc arguments increase number cells factor decrease probability success finally design ldc weaker guarantees use arguments prove lower bounds space weak ldc deterministic data structure definition randomized algorithm gns problem two algorithm specified following three components data structure preprocesses dataset consisting points well order produce data structure depends query chooses two indices specifies function output given require whenever unique neighbor note indices generates probe data structure well function independent definition define following distributions let distribution datasets given sampling times space uniformly random let uniform distribution let distribution queries given first picking dataset point uniformly random picking uniformly random lemma assume randomized algorithm gns using two exists deterministic algorithm gns using two also produces data structure query chooses two indices independently probe well function proof following direct application yao principle success probability algorithm assumption exists distribution algorithms achieve probability success least single query therefore fixed distributions exists deterministic algorithm achieving least success probability order simplify notation algorithm let denote output algorithm write assume outputs pair indices well function algorithm outputs fixed dataset definition allows succinctly state probability correctness query neighbor without caring specific cells algorithm probes function algorithm uses make decision important thing note contents data structure may depend dataset however algorithm produces well indexes probes query point deterministic assume existence deterministic algorithm success probability least using cells width success probability taken random choice dataset making data structures let set queries probe cell probe algorithm sets well defined independently dataset particular could write probes cell probe querying running probing portion algorithm without need specify dataset could write simply trying every query point seeing cells algorithm probes words since algorithm deterministic probing portion algorithm completely specified two collections well function two partitions query space query make probe cell output value observing contents cells define notion data structures informally requires data structure rely one particular cell much namely large definition deterministic algorithm using cells low contention every set use following lemma argue small increase space data structure made lemma suppose deterministic algorithm gns two using cells exists deterministic algorithm gns two cellprobes using cells succeeds probability low contention proof first handle suppose partition enough parts size one set measure part partition make new cell contents cell query lies inside probe new cell data structure side cell contents replicated additional cells number cells data structure since cells size original cell one cell small measure also keep mind modified sets thus cells procedure second collection cell multiple cells size exactly partition one extra small cell total number cells dividing heavy cells second probe lighter cells second probe added cells order make added cells therefore cells additionally contents cells remain algorithm succeeds probability given lemma assume deterministic algorithm gns two using cells low contention extra factor number cells pushed asymptotic notation datasets shatter fix thought sufficiently small constant definition say partition shatters point operator takes part fixed dataset point refer slack shattering slack corresponds total measure leftover remove arbitrary subset measure least lemma shattering let collection disjoint subsets measure shattered remainder section let interested shattering dataset points respect collections dataset points get shattered probe many cells data structure intuitively bit corresponding dataset point stored across various cells point define subsets hold slack shattering respect definition let dataset point shattered let arbitrary subsets satisfies since shattered pick measure refer given collection let event collection shatters note lemma implies lemma high probability choice point dataset points satisfy proof simple chernoff bound expected number points satisfy therefore probability points satisfy exp call dataset good dataset points shattered lemma exists good dataset proof follows via simple argument fixed dataset let simplify notation good good good therefore exists dataset shattered corrupting cell contents shattered points rest proof fix dataset satisfying conditions lemma introduce notion corruption data structure cells parallels notion noise codes remember fixing algorithm produces data structure definition call corrupted version cells differ cells section show exist dataset points set size good recovery probability even algorithm access corrupted version data structure definition fixed let note definitions lemma fix vector let data structure algorithm produces dataset let corruption cells every events occur proof note represents probability mass queries neighborhood algorithm returns want understand much probability mass remove avoid probing corrupted cells since dataset point shattered probability mass come slack shattering detail probability algorithm returns query worst case every query returns thus removing removed probability mass queries algorithm returns correctly remaining probability mass distributed across various cells cell mass probing first probe mass probe second probe therefore remove cells first second probe probe cells probability avoid corrupted cells algorithm output uncorrupted data structure therefore probability mass returns query corrupted data structure least lemma fix small enough constant exists set size whenever events occur taken small constant like proof dataset points shattered simplifying notation let need show since set argument averaging argument combine lemma lemma obtain following condition dataset lemma fix small enough exists set whenever algorithm probes corrupted version data structure proof consider set satisfying conditions lemma whenever gets shattered average recover probability probing data structure input therefore lemma probes corruption cells recover probability least words averaged theorem exists algorithm subset size makes cell probes furthermore corruption cells recover probability least random choice proof order extract generate random query probe data structure cells assuming data structure uncorrupted lemma exists set size algorithm recovers probability least probability taken average possible fix algorithm subset satisfying conditions theorem since fixed dataset satisfying conditions lemma say input algorithm order initialize data structure dataset bit associated decreasing word size reduce case word size bit lemma exists deterministic algorithm input builds data structure using cells width bit well corruption positions satisfies proof given algorithm constructs data structure input construct following data structure cell make cells contain parities bits blows size data structure fix algorithm produces function succeeds probability least exists signed parity input bits equals least inputs let parity bits cell parity bits cell let denote parity negation parity equals possible input strings algorithm evaluate cell containing parity bits cell parity bits cell let isj isk indices cells since find function fixed two cell probes corrupted version algorithm satisfies whenever remainder section prove version theorem algorithms words given lemma modify space probability obtain answer remainder section assume algorithm bit words connecting codes complete proof theorem remains prove following lemma lemma let deterministic algorithm makes cell probes data structure cells width bit handle corruptions recover probability random input whenever fixed size following must hold log proof lemma uses relies heavily notions quantum computing particular quantum information theory applied ldc lower bounds crash course quantum computing introduce concepts quantum computing necessary subsequent arguments qubit vector write qubit linear combination basis states qubit written refer amplitudes system vector tensor product dimension basis states correspond length write basis state binary representation write quantum state given linear combination basis states shorthand corresponds conjugate transpose quantum state mixed state probability distribution quantum states case quantum system state probability represent mixed states density matrix measurement given family positive operators sum identity operator given quantum state measurement corresponding family tors measurement yields outcome probability kmi results state kmi norm norm say measurement makes observation finally quantum algorithm makes query starting state returning one think control qubit taking values state remains unchanged query state receives amplitude queries may made superposition state state becomes weak quantum random access codes gns algorithms definition exists randomized decoding algorithm making queries string algorithm recover two queries probability least paper prove following result ldcs theorem theorem proof theorem proceeds follows show construct quantumldc classical ldc constructs quantum random access code encodes strings log qubits apply quantum information theory lower bound due nayak theorem theorem stated nayak encoding strings states quantum algorithm given query access decode fixed probability least must hold proof follow pattern similar proof theorem assume existence gns algorithm builds data structure think length binary string encoding particular let jth bit algorithm theorem satisfy strong properties ldc preventing applying directly however guarantees particular support corruptions ldc language means tolerate noise rate additionally necessarily recover every coordinate recover also success probability random choice arguments weaker setting random choice proof follows adapting constant let data structure satisfying hypothesis lemma exists quantum algorithm lemma let starting log state copies recover probability random choice assuming lemma complete proof lemma proof lemma proof similar proof theorem let represent system consisting copies state log encoding using lemma assume quantum algorithm given recover probability random choice let von neumann entropy conditional entropy mutual information let system system corresponds uniform superposition strings concatenated encoding let first subsystem corresponding first qubits second subsystem corresponding qubits since qubits therefore mutual information note fano inequality using fact fano inequality works even recover probability averaged additionally therefore furthermore since log log remains prove lemma proceed rest section first show simulate gns algorithm quantum algorithm lemma fix let data structure produced algorithm input suppose exists quantum algorithm makes one quantum query succeeds probability output proof use procedure lemma determine output algorithm input index procedure simulates two classical queries one quantum query quantum algorithms make specified following manner quantum state queries querying resulting quantum state also quantum measurement algorithm obtains state performs measurement algorithm observes outputs algorithm observes outputs lemma know must exist state algorithm succeeds probability probability random quantum algorithm succeeds random order simplify notation write probability making observation state since positive matrix exactly way remove parts quantum state let obtain quantum state keeping amplitudes lemma fix quantum state satisfies proof note since simulate succeed probability least random simplify say since contains parts probes corrupted values algorithm still succeed probability random inputs therefore following two inequalities hold note one verify averaging two inequalities get desired expression lemma fix exists quantum algorithm starting quantum state recover value probability random proof algorithm argument almost identical theorem check works weaker assumptions let lemma know random assume order simplify notation let quantum algorithm starting state want recover probability since otherwise argument work flipped also assume since otherwise simply outputting observation observation work algorithm works following way outputs probability otherwise makes measurement state observation made algorithm outputs otherwise outputs probability success random input returns correctly returns returns probability algorithm returns correctly probability algorithm returns correctly simplifying returns correctly finally complete proof lemma proof lemma proof exactly finishing arguments theorem simply check weaker conditions give desired outcome input access copies state algorithm applies measurement measurement designed order yield state makes observation measurement fact amplitudes large makes valid measurement probability observing used algorithm repeatedly applies measurement observing outcome never makes observation algorithm outputs uniformly random algorithm observe runs output algorithm lemma following simple calculation done gives desired probability success random input returns correctly adaptivity extend lower bounds adaptive setting lemma exists deterministic data structure makes two queries adaptively succeeds probability least exists deterministic data structure makes two queries succeeds probability least proof algorithm guesses outcome first cell probe simulates adaptive algorithm guess knowing two probes make probe data structure algorithm guessed contents first correctly output value algorithm otherwise output random value algorithm succeeds probability least applying theorem adaptive algorithm succeeding probability obtain algorithm succeeds probability intended value lower reduction weak ldc still goes let another consequence one easily verify small enough log yields tight lower bounds factors hamming space log case hamming space compute robust expansion similar fashion theorem particular log let log log require log log log log acknowledgments would like thank jop helping navigate literature ldcs thank omri weinstein useful discussions references nir ailon bernard chazelle fast transform approximate nearest neighbors siam alexandr andoni dorian croitoru mihai hardness nearest neighbor proceedings symposium foundations computer science focs pages alexandr 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workshop computational learning theory pages gregory valiant finding correlations subquadratic time applications learning parities closest pair problem acm previously focs jun wang wei liu sanjiv kumar chang learning hash indexing big data survey available http jingdong wang heng tao shen jingkuan song jianqiu hashing similarity search survey corr yitong yin simple lower bounds approximate isoperimetric inequalities corr zeyuan allen zhu yang yuan karthik sridharan exploiting structure stochastic gradient methods using raw clusters corr random instances first introduce equivalent notion random instance section instance lies core optimal lsh points queries lie unit sphere dataset generated sampling unit vectors independently uniformly random query generated first choosing dataset point uniformly random choosing uniformly random points within distance goal data structure preprocess given query generated recover corresponding data point instance must handled data structure fact show reduce instance several random instances without increasing time space complexity polynomial factor resulting instances exactly random instance described roughly distribution distances data points following strategy first analyze random instance reduce case general subsets instances spherical case describe solve random instance ann unit sphere near neighbors planted within distance defined appendix obtain space tradeoff namely appendix extend algorithm entire space using techniques assume log data structure description data structure single rooted tree consisting levels zeroth level holds root node level children leaves every node let set nodes path root except root node except root holds random gaussian vector stored node define subset dataset parameter chosen later instance since intuitively set corresponds subset dataset lies intersection sphere caps centered around every leaf tree stores subset process query start root make way tree consider children root hzv parameter chosen later recurse end leaf try points find near neighbor end leaf find neighbor fail analysis first let analyze probability success let near neighbor lemma probability successfully finding query least proof prove induction suppose querying algorithm node would like prove conditions lemma probability success leaf statement obvious suppose true children node failure child let understand much space data structure occupies lemma arbitrary point lemma expected space consumption data structure proof total space tree nodes occupy time every point participates average leaves hence desired bound finally let analyze expected query time arbitrary point lemma expected query time proof first query touches tree nodes average node leaf time spent fixed leaf fixed dataset point probability end leaf together query point hence obtain desired bound setting parameters first set log second set log simply substitute parameter setting lemma lemma gives expected space expected query time discussed lemma setting probability success order get desired tradeoff vary suppose want space let log largest number every log substituting values lemma query time follows standard computation given log log computation relatively standard see verify next resulting tradeoff obtained denote real numbers lemma suppose every namely one proof using spherical symmetry gaussians reduce computation computing gaussian measure following set squared distance zero set result follows appendix discussion conclude one achieve following space query time show equivalent tradeoff indeed squaring replacing get simplifying equation get remember hence obtain multiplying similar fashion squaring obtain equivalently squaring obtain simplifying observe obtain equation hence done proving equivalent upper bound general case show extend result appendix general case using techniques particular show reduce instance several instances overall algorithm gives data structure solves ann problem euclidean space using space query time satisfy data structure decision tree however several notable differences whole data structure single decision tree consider collection trees instead spherical lsh used use partitioning procedure section one proceeds partitioning dataset parts contain less points change stopping criterion slightly ensure number branch unlike use property random space partition analysis related fact probability success single tree constant unlike polynomially small reduce general case bounded ball case using lsh quite since aiming getting full instead use standard trick imposing randomly shifted grid reduces log invoke upper arbitrary dataset dataset diameter bound together reduction case enough proceed think log section figure covering spherical cap radius overview start overview consider dataset points assume rescaling may also assume dataset lies euclidean space dimension log log one always reduce dimension applying lemma incurring distortion log log high probability simplicity suppose entire dataset query lie sphere radius done case corresponds random instance points apply data structure section suppose split number disjoint components dense components termed one component termed properties components follows dense component require covered spherical cap radius see fig small quantities chosen later component contains dense components inside proceed separately enclose every dense component slightly smaller ball radius see figure simplicity let first ignore fact necessarily lie boundary enclose dense cluster smaller ball recurse resulting spherical instance radius treat pseudorandom part described section sample gaussian vectors parameter chosen later remainder separately form subsets follows pei hzi parameter chosen later remainder separately recurse pei note recurse may appear new dense clusters sets pei since may become easier satisfy minimum size constraint project figure definition project query procedure recursively query query point component identify hzi query corresponding children recursively parameter chosen later remainder separately analyze algorithm show make progress two ways first dense clusters reduce radius sphere factor hence iterations must arrive case easy argued second random component argue points lie distance particular ratio typical distance exactly like random case reason call setting data structure section performs well address issue deferred description namely dense component generally lie rather occupy interior case partitioning thin annuli carefully chosen width treat annulus sphere discretization ball adds complexity analysis fundamental conceptual point view formal description ready describe data structure formally depends small positive parameters well integer parameter log also need choose parameters remainder separately preprocessing preprocessing algorithm consists following functions processsphere builds data structure dataset lies sphere assuming need solve ann distance thresholds moreover guaranteed queries lie parameter counter sense measures far done processball builds data structure dataset lies inside ball assuming need solve ann distance thresholds unlike processsphere queries arbitrary parameter meaning process builds data structure dataset solve general project auxiliary function computing following projection suppose two spheres common center radii suppose points project returns distance point lies closest see figure elaborate algorithms functions processsphere function processsphere follows exposition section consider three base cases first stop store whole second goal achieved trivially since point works answer valid query third algorithm section would give desired point particular choose appropriately particular set one log make single step otherwise find dense clusters smaller balls radius centers contain many data points least balls enclosed balls invoke balls unconstrained center radius processball remaining points perform single step algorithm section appropriate particular set distance recurse part increased processball first consider following simple base case point could serve valid answer query general reduce spherical case via discretization ball first round distances multiple change distance pair points triangle inequality every possible distance data point every possible distance query admissible integers build separate data structure via processsphere also need check ensure corresponding pair yield trivial instance compute new distance thresholds data structure follows rounding new thresholds ball instance since distances change compute final thresholds projecting query sphere radius invoke project see definition process process reduces general case ball case proceed similarly processsphere two modifications first apply randomized partition using cubes side log solve part separately second seek find dense clusters raoc dius clusters apply reduction case algorithm single iteration algorithm section project implemented formula see figure overall preprocessing creates decision tree nodes correspond procedures processsphere processball process refer tree nodes correspondingly using labels description query algorithm query algorithm consider query point run query decision tree starting root applying following algorithms depending label nodes process first recursively query data structures corresponding clusters second locate spherical caps query data structure built corresponding subsets processball first consider base case return stored point close enough general check return otherwise round distance multiple next enumerate distances potential near neighbor looking query corresponding processsphere children projecting sphere tentative near neighbor using naturally project processsphere proceed exactly way process modulo base cases cases try points store explicitly happens set parameters briefly state one sets parameters data structure recall dimension log log log set follows log log log exp log log log exp sufficiently large positive constant need specify set remainder idea set log time every distance finally choose parameter governs memory consumption gives unique value governs query time crucial relation parameters much smaller log implies large distance effectively equal least sake single step random partition
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scalable magnetic field slam using gaussian process maps manon kok arno university cambridge united kingdom aalto university finland apr abstract present method scalable fully magnetic field simultaneous localisation mapping slam using local anomalies magnetic field source position information anomalies due presence ferromagnetic material structure buildings objects furniture represent magnetic field map using gaussian process model take physical properties magnetic field account build local magnetic field maps using hexagonal block tiling make approach computationally tractable use gaussian process regression combination particle filter show possible obtain accurate position orientation estimates using measurements smartphone approach provides scalable magnetic slam algorithm terms computational complexity map storage introduction use magnetic field source position information indoor navigation promising novel approach gained interest recent years relies upon spatial variation ambient magnetic field typically due ferromagnetic material structures buildings lesser extent due presence furniture advantage using magnetic field positioning measured small device without additional infrastructure without requirements furthermore magnetometers nowadays present almost smartphone crucial approach ability build maps magnetic field used interpolation extrapolation ability localise inside map paper present scalable fully simultaneous localisation mapping slam approach builds map magnetic field simultaneously localising sensor map extending previous work build magnetic field map using gaussian process regression incorporate physical knowledge magnetic field known maxwell equations gaussian process prior overcome issues computational complexity use approach first presented results representation fits perfectly particle filter used building map localisation map resulting tractable localisation algorithm example results found fig approach presented relies basis function expansion specific domain number basis functions needed scales size domain allow mapping large areas propose map representation using hexagonal block tiling aims providing compact terms required storage per volume representation magnetic field map three vector field components associated uncertainties using approach hexagonal tiles created whenever particles enter unexplored areas resulting growing number local gaussian process maps present experimental results obtain magnetometer measurements well odometry smartphone odometry gives information change position orientation resulting pose estimates drift time drift corrected making use magnetometer measurements magnetic field map show accurate pose map estimates obtained large scale experiments illustrated fig previous work magnetic field slam typically assumes motion best authors knowledge first fully approach magnetic field slam also first time approach used slam domain approach chosen hexagonal figure illustration magnetic field map built slam approch hexagonal block radius height cover gaussian process magnetic map volume state dimension detailed map provided fig problem formulation interested estimating position orientation sensor building map magnetic field denote position sensor time instance superscript refers world frame slam formulation always possible move rotate estimated map simultaneously moving rotating position orientation estimates words absolute position orientation observable without loss generality choose origin world frame equal initial position sensor axes aligned coordinate system defined initial orientation orientation sensor denoted qwb refers body frame origin lies centre magnetometer triad axes aligned axes magnetometer encode orientation using unit quaternion direct mapping quaternion qwb use interchangeably double superscript rotation matrix rwb denotes rotation body frame world frame reverse rotation denoted hence rwb denote estimate magnetic field map details exact form found section note assume map magnetic field change time subscript refers fact belief map changes time data results information map resulting state vector given qwb dynamics position orientation modelled terms change position orientation obtained odometry information consider inputs dynamic model using assumption magnetic field vary time dynamic model written expq qwb assume noise position orientation gaussian possibly time subsequent samples denotes quaternion multiplication expq refers representation noise vector unit quaternion dynamics orientation qwb interpreted rotation equivalent dynamics position information quaternion algebra representations orientation see use magnetometer measurements provide information magnetic field map well position orientation sensor modelled rbw prediction magnetic field location details specific form found section note magnetic field map represented world frame hence magnetometer measurements modelled prediction rotated body frame together prior state first time instance models constitute state space model use rbpf estimate state defined initial position orientation fix world coordinate frame set initial position orientation zero uncertainty magnetic field map first time instance set equal gaussian process prior details gaussian process map discussed section rbpf implementation subsequently discussed section methods gaussian process magnetic field map building previous work build gaussian process magnetic field map encode physical knowledge magnetic field first know measured magnetic field consists earth magnetic field magnetic field due anomalies induced variations building structures furthermore using maxwell equations classical electromagnetism know magnetic field due anomalies modelled using latent scalar potential field spatial coordinate world frame magnetometer measures derivative scalar potential assume scalar potential field realisation gaussian process prior magnetometer measurements corrupted gaussian noise resulting following model observation local earth magnetic field contributes linearly scalar potential magnitude scale hyperparameter magnetic field anomalies captured squared exponential covariance function exp magnitude characteristic hyperparameter including additional prior information introduced scalar potential results coupling three components magnetic field according physical laws shown improve prediction accuracy simply modelling magnetic field component separately using model possible predict magnetic field previously unseen locations practice however quickly becomes computationally intractable large amount magnetometer measurements collect fact computational complexity scales computationally tractable solution presented problem projected eigenbasis negative laplace operator confined domain domain eigendecomposition laplace operator subject dirichlet boundary conditions solved using eigendecomposition possible approximate covariance function sse approximated using basis functions corresponding eigenvalues exact form depends shape domain note sse denotes spectral density function squared exponential kernel sse closed form solution exists see using approximation boundary conditions defined implies model magnetic field reverts back earth magnetic field boundary domain fixed domain desirable slam problem know priori spatial extent final magnetic field map furthermore number basis functions needed good approximation scales size domain large scale slam problems hence become intractable therefore propose scalable representation split magnetic map grid hexagonal blocks radius height subdomain given hexagon center point dth hexagon cell grid representation hexagonal grid provides efficient way maximising area per number basis functions representing magnetic field map thus requiring less memory storing map representation compute basis functions corresponding eigenvalues need solve eigendecomposition laplace operator hexagon solved closed form instead solved numerically set eigenvalue problem composing sparse stencil matrix corresponding laplacian using finite difference approximation solution given lanczos algorithm see callable matlab eigs use solving largest real eigenvalues corresponding eigenfunctions hex fig shows first eigenfunctions negative laplacian unit hexagonal domain respect dirichlet boundary conditions eigendecomposition extended cover hexagonal grid cell considering numerically solved horizontal eigendecomposition solution separable vertical dimension final eigenbasis becomes combination sin hex matrix consists index set permutations integers corresponding largest eigenvalues numerical solution hexagon fast takes less second even tight discretisation average laptop computer furthermore basis functions independent hyperparameters input locations need evaluated causing computational overhead slam algorithm basis function expansion described allows write prior model hexagonal tile time terms mean covariance diag sse sse sse posterior updated sequentially new magnetometer data arrives written rbw figure first eigenfunctions negative laplace operator hexagonal domain subject dirichlet boundary conditions eigenbasis effectively solved numerical solver calculations required note kalman filter measurement update state representing magnetic field map specific tile fact exploited magnetic field slam algorithm presented section magnetic field slam discussed section interested estimating position orientation sensor building map magnetic field represent map terms varying number hexagonal block tiles using basis function expansion presented section use nonlinear filtering technique estimate position orientation map tile visited state vector relatively large dimension due representation magnetic field map instance section set dimension state per hexagonal block however enters conditionally linearly state space model possible exploit using particle filter rbpf uses kalman filter conditionally linear states xlt particle filter xnt nonlinear states xnt qwb xlt number hexagonal block tiles created time rbpf particles position orientation used represent state time particles also contains estimate map covariance note consists magnetic field map hexagonal block particle visited assume tile independent covariance linear state therefore described dynamics state chosen measurement model given important thing note discussed section absolute position orientation observable orientation particles set initial position equal initial position orientation qwb initialise magnetic field according gaussian process prior see also section state space model presented section results fairly straightforward rbpf implementation major difference standard rbpf particle updates local magnetic field map current location note computational complexity algorithm scales linearly number particles linearly number hexagonal tiles per particle alg magnetic field slam input output particle initialisation initialise particles qwb set given initialise weights particles create new hexagonal block tiles new hexagonal block tile create tile particle initialise given evaluate importance weights evaluate importance weights based measurement model normalise resampling particles tile revisiting resample particles replacement kalman filter measurement update update magnetic field map covariance hexagon particle close neighbours using nonlinear states according obtain filtered estimate output position orientation estimated map particle highest weight particle filter time update predict new particles expq drawn respectively set practical magnetic field slam implementation care must taken following terms updating map avoid boundary effects due dirichlet boundary conditions actual tile extends slightly outside hexagonal block domain updating allow smooth transition one tile next particles close border tile also update map neighbouring tiles exploration phase particles start building map previously unseen locations little information available distinguish particles would like ensure particle cloud keeps spreading phase end adapt rbpf implementation two different ways maintain particle spread exploration phase delay updating map one lengthscale collapse particle cloud quickly resample particles revisit tile resampling delayed least particles arrived tile revisit results empirical experiments use data captured smartphone simultaneously building magnetic field map localising use apple iphone published september previous flagship product apple current standards represents standard smartphone interest paper purely map building localisation choose leverage recent developments pedestrian pdr methods use arkit pdr provided apple arkit released september used version running ios using imu camera phone provide position orientation pdr movement meters magnetic magnetic magnetic odometry tilted view figure slam solution walking square several loops around meters subfigures show norm magnetic field map highest weight particle transparency maps scales marginal variance map subfigures show corresponding vector field components final map odometry trajectory visualised shows hexagonal tiles magnetic field maps trajectory arkit visual relocalisation hood aims correct drift resulting discontinuities pdr track discontinuities turned harmful useful approach implemented heuristic filtering approach removing relocalisation jumps surprisingly drift position orientation see fig data captured customised data capture application running phone collects imu magnetometer data arkit pdr estimates furthermore captured data complemented video frames phone camera resolution fps barometric air pressure gnss locations captured phone none used slam implementation captured validation reference data stored device downsampled use slam implementation run mathworks matlab implementation simplicity use model parameters experiments set size hexagonal blocks number basis functions per tile radius height used solving eigenbasis laplacian extends one meter outside hexagon set gaussian process hyperparameters defined section assumes lengthscale magnetic field anomalies order measurement noise rather high also account model discrepancies set number particles rbpf neighbouring tiles updated samples closer hexagonal block boundary process noise parameters set diag values specified meters drift per second diag values specified degrees drift per second terms memory usage choices mean initialised map tile particle state dimension grows fast implementation uses hashmap per particle storing keeping track map tiles tile stores mean covariance matrix illustrating magnetic slam first experiment data set collected dpo engineering department university cambridge significant excitation magnetic field present instance due presence radiators large number computers trajectory around meters long covers area around circular trajectory traversed several times easy visually inspect quality estimates slam algorithm shown fig different subfigures illustrate workings algorithm figs show estimates progress time visualisation purposes norm magnetic field predicted map highest weight particle large number locations hexagon transparency map visualises uncertainty map seen uncertainty large unexplored regions map actually visible uncertainty map decreases time regions black line displays trajectory highest weight particle figs deliberate slight delay updating map clearly seen discussed section wait resampling least particles tile previously updated left resampling done first loop around seconds spread particles fig therefore fairly large allows algorithm properly close loop visualised fig another aspect algorithm note fig particles border tile trajectory started new tile allow smooth transitions two tiles many particles case therefore update local magnetic field maps tiles map visualised figs shows norm predicted magnetic field model however explicitly models magnetic field vector instead visualising norm predicted magnetic field therefore also possible visualise predicted magnetic field three directions shown figs earth magnetic field fairly large cambridge magnetic field significantly larger magnitude based fig concluded slam algorithm estimates trajectory high accuracy since trajectory seen traversed multiple times comparison also show odometry fig clearly shows drift position magnetic slam approach able correct drift illustrate fact even though movement data set plane estimation still done tilted view shown fig fully magnetic slam section movement close hexagonal tiles local magnetic field map created whenever particles border tile exactly procedure also applies movement tile block structures three dimensional space fig show estimated trajectory magnetic field map empirical experiment first walked subsequently set stairs engineering department university cambridge hexagonal block local magnetic field map visualise norm predicted magnetic field hexagonal slice middle block shown transparency visualises uncertainty map results presented section algorithm starts resample particles way stairs seen results estimated path fairly similar directions magnetic slam third empirical data set walk around john college cambridge college consists several courts even though data set captured outdoors close proximity buildings results small magnetic field anomalies used localisation slam results previously presented fig presented detail fig map extends tiles covering courtyard many algorithms extent data set would therefore prohibitively figure example showing nature path covers flight stairs university cambridge engineering department complete path length meters tiles show magnetic map norm color scale subfigure previous figure meters figure slam map solution example visualised color field norm magnetic field opacity follows uncertainty layout corresponds fig large however due computationally efficient model terms hexagonal tiling approximate gaussian process model rbpf algorithm computations remain feasible seen fig approach scales magnetic slam conclusions future work presented scalable algorithm magnetic field slam builds local gaussian process maps map approximation local full solution terms basis function expansion approximation allows map magnetic map volume state dimension local map modelled terms hexagonal block tile minimises number tiles needed cover total mapping volume formulation fits perfectly rbpf resulting computationally tractable algorithm computational complexity scales linearly number particles particle filter number hexagonal tiles particle algorithm shown perform well three challenging empirical data sets one challenges data used testing odometry obtain arkit always obey motion model specifically straight paths odometry shows almost drift fast turns introduce sudden errors follow motion model model constant process noise future work would like focus using inertial pdr odometry instead apart avoiding issues would also result positioning algorithm based widely available inertial magnetometer sensors directions future work direction using particle smoother reducing computational complexity algorithm acknowledgments research financially supported epsrc grant autonomous behaviour learning uncertain world grant number academy finland grant sequential inference probabilistic modelling grant number references janne haverinen anssi kemppainen global indoor based ambient magnetic field robotics autonomous systems binghao thomas gallagher andrew dempster chris rizos feasible use magnetic field alone indoor positioning proceedings international conference indoor positioning indoor navigation ipin pages michael angermann martin frassl marek doniec brian julian patrick robertson characterization indoor magnetic field applications localization mapping proceedings international conference indoor positioning indoor navigation ipin pages etienne grand sebastian thrun magnetic field mapping fusion indoor localization ieee conference multisensor fusion integration intelligent systems mfi pages solin kannala rahtu terrain navigation magnetic landscape particle filtering indoor positioning proceedings european navigation conference david hanley alexander faustino scott zelman david degenhardt timothy bretl magpie dataset indoor positioning magnetic anomalies proceedings international conference indoor positioning indoor navigation ipin pages chao gao robert harle msgd scalable indoor magnetic graphslam proceedings ieee international conference robotics automation icra pages david rambla raul montoliu oscar belmonte huerta new database magnetic localization problems proceedings international conference indoor positioning indoor navigation ipin pages solin kok modeling interpolation ambient magnetic field gaussian processes ieee transactions robotics accepted publication carl edward rasmussen christopher williams gaussian processes machine learning mit press arno solin simo hilbert space methods gaussian process regression arxiv preprint thomas fredrik gustafsson nordlund marginalized particle filters mixed models ieee transactions signal processing patrick robertson martin frassl michael angermann marek doniec brian julian maria garcia puyol mohammed khider michael lichtenstern luigi bruno simultaneous localization mapping pedestrians using distortions local magnetic field intensity large indoor environments proceedings international conference indoor positioning indoor navigation ipin pages ieee martin frassl michael angermann michael lichtenstern patrick robertson brian julian marek doniec magnetic maps indoor environments precise localization legged nonlegged locomotion proceedings international conference intelligent robots systems iros pages ilari vallivaara janne haverinen anssi kemppainen juha simultaneous localization mapping using ambient magnetic field proceedings ieee conference multisensor fusion integration intelligent systems mfi pages ilari vallivaara janne haverinen anssi kemppainen juha magnetic slam method solving localization problem mobile robot task proceedings international conference advanced robotics icar pages kok hol using inertial sensors position orientation estimation foundations trends signal processing david jeffrey griffiths reed college introduction electrodynamics prentice hall upper saddle river gene golub charles van loan matrix computations johns hopkins university press edition simo bayesian filtering smoothing cambridge university press
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reduced dynamic mode decomposition christian tim mitchell rin abstract identification models data challenging task even identified system suitable certain data set generally approximate behavior data source work consider dynamic mode decomposition method system identification compare excitation approaches identification process describe stabilization strategy identified systems keywords dynamic mode decomposition model reduction system identification cross gramian optimization msc dec peter benner computational methods systems control theory max planck institute dynamics complex technical systems sandtorstr magdeburg germany orcid benner computational methods systems control theory max planck institute dynamics complex technical systems sandtorstr magdeburg germany orcid himpe computational methods systems control theory max planck institute dynamics complex technical systems sandtorstr magdeburg germany orcid mitchell introduction rin various applications difficult sometimes even impossible derive models first principles however response system data inner state may still available refer setup graybox example natural process whose underlying action well understood considered graybox since may able measure behavior applications manufacturing design may necessary model provided subcomponent whose exact full specifications may obtainable due containing proprietary information order gain insight natural technical processes derive models simulate predict behaviors often beneficial perhaps necessary create models using measured generated data discipline system identification investigates methods task obtaining models data one class methods identification dynamic mode decomposition dmd also provides modal analysis resulting systems work investigate variants dmd class systems compare data sampling strategies dmd roots modal decomposition koopman operator recently rediscovered spectral analysis fluid dynamics basic dmd method introduced various extensions added exact dmd extended dmd furthermore dmd also used tool model order reduction proposed using dmd flow analysis control dmd also combined techniques model nonlinear systems comprehensive survey dmd variants see work builds upon two specific variants dmd first dynamic mode decomposition control dmdc thus relation also koopman inputs control kic second dynamic mode decomposition iodmd closely related direct numerical algorithm state system identification dmdc extends dmd scenarios input discrete system approximation given functional iodmd additionally handles case system output specified also functional generically identify system without prior knowledge relevant input functions techniques persistent excitation well known time propose extension iodmd method new excitation variant related cross gramian matrix additionally since identification guarantee resulting models stable propose procedure stabilize models employing nonsmooth constrained optimization method solve corresponding stabilization problem document structured follows section overview prerequisite dynamic mode decomposition method relevant variants given section describes new excitation strategy optimizationbased stabilization procedure discussed section finally two numerical experiments conducted section dynamic mode decomposition consider ordinary differential equation ode state vector field furthermore consider sampled uniform intervals times basic version dmd aims approximate constructing linear system axk rin linear operator also hold starting initial state sequence defined corresponds trajectory state vectors corresponding data matrix simply concatenation state vectors forming following two partial trajectories data matrix last data point removed first data point removed idea behind plain dmd solve linear system equations fact replaced respectively solution problem given pseudoinverse also solution minimizing error frobenius norm arg min dmd modes given eigenvectors matrix beyond using single time series dmd also generalized method called exact dmd additionally support concatenation multiple different time series time steppings generalization dmd made possible reducing requirements lesser condition columns need composed pairs data fulfilled practical computation give algorithmic description dmd identification pseudoinverse data matrix computed using singular value decomposition svd follows rin however inverting tiny nonzero singular values computed diagonal matrix poses numerical problem small singular values may inaccurate applying could overamplify components particular less important ones counteract effect computing pseudoinverse via svd done practice truncating singular values smaller fixed equivalent adding regularization term solution arg min value regularization parameter following algorithm computation system matrix given trajectory lower bound discarding tiny singular values follows sample trajectory form data matrix assemble shifted partitions compute svd otherwise truncate identify approximate discrete system matrix dmd variant order dimension computed matrix equal number retained nontruncated singular values truncation done solely numerical accuracy intention keep many components dynamics possible contrast model order reduction typically aims significantly reduce system small set essential dynamics accomplishing goal focus section dynamic mode decomposition control considering systems whose vector field depends state also input function led dmd variant called dynamic mode decomposition control dmdc focus specific dmdc variant sec aims approximate linear control system axk buk called input operator must also identified addition input version continuous input function sampling times given addition forming plain dmd section analogous construction matrices input data also done concatenation series input data vectors done obtain matrix partial data matrix simply without last column last input sample rin obtained computing approximate solutions linear system equations given replaced solution mentioned section dmdc actually special case kic method kic state system also augmented discretized input leads resulting augmented system additional operators course two additional operators must also identified along matrices however one assumes input known identification associated operators required zero matrices thus clear kic generalization dmdc dynamic mode decomposition extending underlying system also include output function associated output functional also depends state input yields following system modeling systems form given given rise class dmd methods called dynamic mode decomposition iodmd similar previously discussed dmd variants iodmd method also approximates original system linear system input output axk buk cxk duk output operators respectively since approximation includes output data discrete output instances also correspondingly arranged matrix concatenation simply omits last column output sample rin matrices approximated solving equivalently replaced respectively solution procedure equivalent direct algorithm mentioned section note dmd variants discussed far identify original continuoustime systems linear models however one create continuousb approximation time model given model obtained iodmd using example standard firstorder euler method hbu axk buk hax output operators model remain model produced iodmd finally important note dmd derived models generally valid time horizon data gathered reduced order dmd accelerate computation models follow reduce order possibly state trajectories using approach data matrices compressed using truncated galerkin projection order identified system thus determined rank projection popular method compute truncating projection proper orthogonal decomposition pod practically obtained left singular vectors pod modes data matrix svd rin number resulting computed modes depends desired proppod jection error xkf dii consequence lemma see example note data compression scheme different motivation compared aforementioned done computing pseudoinverse via truncated svd discussed section latter truncation based magnitude singular values done reasons numerical accuracy computing pseudoinverse applying subsequent computations truncation discussed using sum singular values squared allows possibility much less onerous computational burden state space models often greatly reduced discarding handful essential modes order obtain desired approximation error datadriven model reduction techniques compression state trajectory similarly applicable example empirical balanced truncation training data generic identification dmd method hence source data used system identification needs considered usually possible identify inputoutput system provided discrete input state output functions identified system produces similar outputs given input used identification identify model data associated system needs produce outputs approximating data source specific data sets whole class relevant input functions perhaps even arbitrarily admissible ones generic linear system identification matrices computed manner require specific data set allows behaviors system modeled predicted given initial condition input function accomplished example persistent excitation utilizes signals step functions gaussian noise training input data propose related approach also exploits random gaussian sampling yet based cross operator cross excitation cross operator tool model reduction encodes coherence associated system linear systems cross gramian matrix operator defined composition controllability operator observability operator thus square system input output space dimension maps given initial state via observability operator output function turn passed controllability operator input function resulting final rin generate trajectory data modify cross operator replacing controllability operator map yields operator maps initial state output function maps output input function state trajectory instead final state compared cross excitation procedure using perturbations initial state generate excitation opposed perturbing input directly cross excitation related indirect identification systems also process first intermediary system openloop system identified used second step generate signal acts excitation identification actual closedloop system distinct difference compared indirect identification approach latter exclusively uses data former also uses state trajectory data addition data stabilization dmd variants methods iodmd included typically desired preserve stability reduced identified systems however models derived iodmd guaranteed stable enforce stability additional step required example proposed stabilizing models derived using projections solving sequence semidefinite programs paper take much direct approach square matrix stable spectral radius less one max although spectral radius nonconvex continuous function respect matrix furthermore continuously differentiable almost everywhere mathematical sense words set points spectral radius nonsmooth measure holds points chosen randomly probability outside set hence despite also illustrated fig nonsmoothness spectral radius still possible attain wealth information gradient since defined subset measure full space thus matrix model stable could consider employing optimization method stabilize hopefully ensuring resulting stabilized version iodmd solution still models original system order meet two objectives consider solving following constrained optimization problem arg min rin margin stability tolerance unstabilized iodmd model already solution solving promote solutions still close original iodmd model simultaneously enforcing requirement models must stable due presence stability radius inequality constraint furthermore unstabilized iodmd model make good point start optimization method however difficulties solving iteratively via optimization techniques first objective function typically underdetermined dmd settings adversely impact method usual rate convergence minimizers longer locally unique however goal mainly stabilize iodmd model without changing properties much necessarily need solve exactly iterations may needed accomplish task alternative one could consider solving admd bdmd min cdmd ddmd lieu admd bdmd cdmd ddmd matrices produced iodmd upside helps avoid problem underdeterminedness arising encourages stable solution close original system found however modified objective longer takes observed data account evaluate alternate optimization problem experiments performance models produced sometimes worse report results experiments done using section second issue trying solve spectral radius rather difficult function optimize relatively speaking first despite continuously differentiable almost everywhere spectral radius still nonsmooth function specifically matrices multiple eigenvalues attain maximum modulus value spectral radius typically minimizers spectral radius matrices example see plots spectral configurations post optimization section appendix optimizing spectral radius often means optimization method must try converge nonsmooth minimizer difficult prospect worse still spectral radius also lipschitz matrices multiple eigenvalues attaining value spectral radius coincide see many available continuous optimization methods rin designed assumption functions optimize smooth least locally lipschitz functions optimized meet criteria methods typically break furthermore nonconvexity spectral radius means optimization codes may get stuck local minima may may provide sufficiently acceptable solutions although inclusion spectral radius constraint makes nonsmooth nonconvex optimization problem constraint function necessarily need solve exactly though remain seen experiments furthermore much literature nonsmooth optimization historically focused unconstrained problems also recent interest addressing problems nonsmooth constraints example new algorithm combining bfgs updating sqp sequential quadratic programming recently proposed general nonsmooth nonconvex constrained optimization problems special knowledge structure assumed objective constraint functions particularly relevant approach method evaluated different spectral radius constrained optimization problems promising results relative solvers section indicates may also good solver nonsmooth constrained optimization problem thus propose using solve specifically use granso algorithm optimization matlab code implementing aforementioned method numerical results implemented new iodmd variant using sampling strategies presented section collect observations original system behaviors furthermore software also optionally stabilize resulting models using granso associated nonsmooth constrained optimization problem discussed section underdetermined optimization problem dmd settings since problems sense quite flat norm gradient merely small poor measure terminate correspondingly tightened granso default termination tolerance relatedly convergence also slow choice starting point also critical goal specified stabilize model minimizing tradeoff increased approximation error may occur due stabilization simply used unstable models starting points granso used granso custom termination feature halt optimization model found stable used objective value less times objective function evaluated original unstable model found loose limit much objective function allowed increase adequate retain good output errors ran granso using bfgs mode good introductory reference many optimization techniques referred paper see default behavior recommended choice nonsmooth problems kept granso options default values well numerical experiments implemented matlab language run matlab workstation computer intel core cores ghz memory excitation stabilization evaluation rin demonstrate effects different types excitation used iodmdbased system identification numerical example specifically given target data set identify linear system first using target data second persistent excitation third utilizing herein proposed cross excitation section considered system based transport equation left boundary domain selected input right boundary output partial differential equation discretized space using upwind scheme spatial resolution resulting ode system given target data given discrete input state output functions simulation performed first order implicit method used width time horizon type input function comparison linear system first identified snapshots generated simulation excited input obtain baseline modeling performance iodmd variant sampled using gaussian noise unit step function input used identification finally variant initial state sampled unit gaussian distribution shifted initial state tested methods tested iodmd also stabilization iodmd without stabilization fig depicts relative output error simulations systems identified using data associated target input increasingly accurate data compression increasingly smaller amounts compression data dimensionality reduced using pod method prescribed projection errors set experiments use stabilization output error output error target projection error identification noise input randomly sampled initial state target projection error identification step input shifted initial state output error output error target rin projection error target regularized identification noise input randomly sampled initial state projection error regularized identification step input shifted initial state figure first numerical experiment section iodmdbased system identification plots show identified reduced order system output error compared original system output varying accuracies pod model reduction figs iodmd procedure regularized truncating singular values regularization used figs figs system identification performed using gaussian noise initial state sampled gaussian distribution respectively figs respectively identification driven step input shifted initial state set experiments using target data produced stable systems large values acceptable projection error models always unstable thus poor performance regardless projection error contrast method produced stable systems levels projection error tested comparing models also happened stable models errors less one order magnitude higher see figs hand using step input shifted initial state produced models increasing accuracy level acceptable projection error data decreased seen figs fig see regularization limited attainable accuracy system constant error independent projection error data iodmd stabilization rin second set experiments fig still shows relative output error simulations systems identified using target data increasingly accurate data compression systems using approach enforce stability necessary data dimensionality reduced using pod method prescribed projection errors subfigures arranged fig step function shifted initial state unaffected stabilization phase systems already stable thus plots figs figs case using gaussian noise randomly sampled initial state figs yielded stable systems target data either without regularization procedure enforced stability models unstable granso able stabilize average number iterations needed find first stabilized version maximum furthermore problems full termination criteria met less iterations average maximum iteration counts problems respectively demonstrates approach indeed able stabilize models reliably efficiently solving via granso also met secondary goal stabilization achieved without deviating significantly original unstable models largest observed relative change initial unstable model corresponding stabilized version average observed relative change merely relative differences calculated comparing vec matrices granso computed stabilized solution similar vec original unstable model reduced orders runtimes compare order identified systems order identified system determined projection error selected compression pod data set target gaussian noise gaussian sample step input shifted initial state resulting reduced order plotted different prescribed projection errors fig see method behave similar terms system dimension variant resulted smaller system dimensions output error output error target projection error identification noise input randomly sampled initial state target projection error identification step input shifted initial state output error output error target rin projection error target regularized identification noise input randomly sampled initial state projection error regularized identification step input shifted initial state figure second numerical experiment section stabilized system identification plots show identified reduced order system output error compared original system output varying accuracies pod model reduction terms computational cost target variants required stabilization see increased runtimes shown red green plots respectively fig otherwise runtimes mostly identical variants comparison bfgs stabilization one potential downside stabilization procedure bfgs inverse hessian updating granso recommended setting nonsmooth problems requires work storage quadratic number optimization variables number system dimension stabilization runtime target gauss gauss step shift projection error projection error rin target gauss gauss step shift comparison identified system reduced order different mean projection errors used experiments see section comparison runtimes seconds stabilization procedure second experiment section figure comparison reduced identified system orders stabilization runtimes tion variables running time required solve could become unacceptably long reduced order model increased thus also consider whether bfgs updating also effective solving using bfgs benefit work storage reduced linear amount number optimization variables however one tradeoffs convergence quite slow practice smooth problems bfgs converges superlinearly bfgs linearly nonsmooth problems linear convergence best one typically expect another potential issue much evidence supporting bfgs reliable nonsmooth optimization case using bfgs nonsmooth problems much less clear good overview literature topic see section investigate question reran experiments section second time granso bfgs mode enabled specifically configured granso approximate inverse hessian iteration using recently computed gradients accomplished setting parameters experimental setup kept described earlier configuration granso often required significantly iterations one might expect average max number iterations find first stable version model respectively order magnitude iterations incurred using bfgs hand problems stable models encountered within first iterations meet full termination criteria average max number iterations respectively roughly two half times incurred using bfgs rin nevertheless problems still satisfactorily solved less iterations matching earlier result using bfgs despite large increases iteration counts granso overall runtime average times faster enabling bfgs respect output error evaluation resulting stabilized models still essentially matched earlier results using bfgs one notable exception target data using smallest projection error granso remarkably found model using bfgs however observe quality stabilized models appeared much sensitive changing granso parameters using bfgs consequence still advocate solving granso generally best done using default bfgs updating nonetheless simply feasible computationally one may still able obtain good results using bfgs perhaps reliably consistently final clarifying remark topic note one necessarily expect good performance nonsmooth problems using optimization code generally critical choice software one specifically designed nonsmooth optimization indeed highlighted evaluation done section codes built smooth optimization perform much worse test set nonsmooth optimization problems compared codes specifically built nonsmooth optimization mind conclusion work evaluated approximation quality iodmd system identification using novel excitation scheme new postprocessing procedure ensure stability identified systems new cross excitation strategy particularly used random sampling often produces better results using persistent excitation experiments indicate excitation schemes useful efficiently obtaining good models approximating target data furthermore show directly solving nonsmooth constrained optimization problem indeed viable approach stabilizing systems retaining salient properties approximating output response code availability source code presented numerical examples obtained http authored christian himpe tim mitchell references alla kutz nonlinear model order reduction via dynamic model decomposition siam sci doi amsallem farhat stabilization reducedorder models numerical methods engineering rin annoni gebraad seiler wind farm flow modeling using inputoutput dynamic mode decomposition american control conference acc pages annoni seiler method construct parametervarying models international journal robust nonlinear control antoulas approximation dynamical systems volume adv des control siam publications philadelphia eykhoff system identification survey automatica brunton johnson ojemann kutz extracting coherent patterns neural recordings using dynamic mode decomposition journal neuroscience methods burke overton variational analysis spectral functions math ser doi chen rowley variants dynamic mode decomposition boundary condition koopman fourier analyses nonlinear science curtis mitchell overton method nonsmooth nonconvex constrained optimization evaluation using relative minimization profiles optim methods fernando nicholson structure balanced principal representations siso systems ieee trans autom control holmes lumley berkooz rowley turbulence coherent structures dynamical systems symmetry cambridge monographs mechanics cambridge university press cambridge doi ionescu fujimoto scherpen cross operator singular value analysis nonlinear symmetric systems proc eur control pages url http katayama subspace methods system identification communications control engineering springer london koopman hamiltonian systems transformation hilbert space proceedings national academy sciences url http rin kutz brunton brunton proctor dynamic mode decomposition modeling complex systems society industrial applied mathematics philadelphia usa doi lall marsden empirical model reduction controlled nonlinear systems proc ifac world congress volume pages url lewis overton nonsmooth optimization via methods math ser mezic spectral properties dynamical systems model reduction decompositions nonlinear dynamics mitchell granso algorithm optimization http see also nocedal wright numerical optimization springer new york oku fujii direct subspace model identification lti systems operating ieee conference decision control pages proctor brunton kutz dynamic mode decomposition control siam applied dynamical systems proctor brunton kutz generalizing koopman theory allow inputs control arxiv cornell university url https rowley dawson model reduction flow analysis control annual review fluid mechanics doi rowley mezic bagheri schlatter henningson spectral analysis nonlinear flows journal fluid mechanics schmid dynamic mode decomposition numerical experimental data fluid rowley luchtenburg brunton kutz dynamic mode decomposition theory applications journal computational dynamics rin van den hof schrama indirect method transfer function estimation closed loop data automatica van overschee moor numerical algorithms state space subspace system identification ifac proceedings volumes volume pages viberg methods identification linear timeinvariant systems automatica
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algorithms synchronization problems compact groups amelia alexander afonso ankur oct department mathematics massachusetts institute technology computer science artificial intelligence lab massachusetts institute technology department mathematics center data science courant institute mathematical sciences new york university usa october abstract various alignment problems arising microscopy community detection time synchronization computer vision fields fall common framework synchronization problems compact groups goal problems estimate unknown vector group elements given noisy relative observations present efficient iterative algorithm solve large class problems allowing compact group measurements multiple frequency channels fourier modes generally irreducible representations group algorithm highly efficient iterative method following blueprint approximate message passing amp recently arisen central technique inference problems structured estimation compressed sensing augment standard ideas amp ideas representation theory algorithm work distributions compact groups using standard methods statistical physics analyze behavior algorithm gaussian noise model identifying phases problem easy computationally hard statistically impossible particular evidence predicts algorithm optimal many cases remaining cases show evidence gaps introduction among common data problems sciences recovering signal present noisy matrix standard tool problems principal component analysis pca estimates signal top eigenvectors one example many macroeconomics large noisy correlation matrices reveal useful volatility yield predictions top eigenvectors however many particular applications involve extra structure sparsity signal structure ignored conventional pca leading estimates thus major topic recent interest first two authors contributed equally ameliaperry work supported part nsf career award grant mit nec corporation email awein research conducted government support awarded dod air force office scientific research national defense science engineering graduate ndseg fellowship cfr email bandeira supported nsf grant part work done department mathematics massachusetts institute technology email moitra work supported part nsf career award grant mit nec corporation google faculty research award email machine learning devise efficient algorithms sparse pca pca general bayesian pca prior variants problems pose major computational challenge significant advances appeared also expected fundamental gaps statistically possible done efficiently thus carried practice large datasets prevalent number recovery problems involve significant amount symmetry group structure compelling example orientation problem microscopy one given many noisy images copies unknown molecule different unknown orientations goal estimate orientations order assemble images estimate molecule structure thus one tasked learning elements one image based loss function derived observed images moreover loss function symmetry depends relative alignments priori reference frame one previous approach problem due produces matrix pairwise image comparisons attempts extract rotations top eigenvectors matrix however reasonable imagine approach could significantly pca exploit significant group structure signal many problems similar patterns group symmetry previously studied general heading synchronization problems general synchronization problem asks recover vector group elements given noisy pairwise measurements relative group elements spectral methods among common techniques used problems singer introduced pca approach angular synchronization analogue problem symmetry one estimates orientations noisy randomly rotated copies unknown image cucuringu applied similar approach sensor localization problem synchronization structure euclidean group euc problem detecting two subcommunities random graph viewed synchronization spectral methods long history use community detection minimum cut problems instances synchronization appear time synchronization networks computer vision optics alignment signals processing work synchronization focused ways better exploit group structure one method used practice related problems alternating minimization alternates estimating rotations aligning images previous guess molecule structure estimating molecule structure images using rotations method appears succeed given strong initial guess molecule structure unclear whether final estimate mainly reflects observations simply initial guess leading problem model bias see paper interested novo estimation without substantial initial guess steering clear pitfall convex relaxations maximum likelihood estimators shown promise semidefinite relaxation angular synchronization introduced singer proven tight bandeira semidefinite programs community detection correlation clustering problems constitute indeed general semidefinite relaxation synchronization problems introduced bandeira performance remains unclear even empirically convex program solved polynomial time large enough make experiments application difficult alternate approach iterative method due boumal following gaussian model angular synchronization one wants estimate vector whose entries complex numbers standing rotations given matrix ratio snr parameter gue matrix independent complex gaussians hermitian symmetry denotes conjugate transpose one could perform ordinary pca initializing small random guess repeatedly assigning method power iteration instead boumal iterate divides entry norm thus projecting unit circle method highly efficient moreover observed produce better estimate pca parameter sufficiently large however pca produces nontrivial estimate projected power method appear produce meaningful estimate somewhat larger fact heuristic analysis similar section suggests required behavior suggests iterative method combining best features pca projected power method might outperform statistically remaining efficient importantly motivated find analogous iterative methods groups complicated observation models one observation model follows instead observing matrix suppose given matrices corresponding different fourier modes denotes entrywise power independent gue matrices approach clear effectively couple information matrices give substantially better estimate could derived one games semidefinite program able use data observation model yet empirically clear performs hope efficient iterative algorithm strongly leverage data multiple frequencies channels question abstract relevance due fourier theory large class measurement models decomposes observations different frequencies manner resembling model moreover analogous statement holds compact groups fourier theory replaced noncommutative setting representation theory thus aforementioned spectral approach applies pca lowest frequency part observations algorithm use frequencies effectively might demonstrate dramatically improved statistical performance paper present iterative algorithm meet challenges algorithm aims solve general formulation synchronization problem apply problems large class observation models symmetry compact group approach statistically powerful empirically providing better estimate pca projected power method synchronization leveraging multiple frequencies give several orders magnitude improvement estimation error experiments see figures indeed conjecture based ideas statistical physics many regimes algorithm statistically optimal providing minimum mean square error mmse estimator asymptotically matrix dimensions become infinite see section finally approach highly efficient iteration taking time linear matrix input roughly iterations sufficing convergence experiments algorithm follows framework approximate message passing amp based belief propagation graphical models related cavity method statistical physics following general blueprint amp algorithms previously derived analyzed compressed sensing sparse pca pca pca projected power methods appeared earlier literature instance planted clique general structured pca fact amp already derived gaussian observation model algorithm generalize one compact groups striking feature amp asymptotic performance captured exactly particular equation called state evolution enabled rigorous understanding performance problems amp provably statistically optimal many cases including gaussian synchronization modulo technicality whereby proof supposes small warmstart amp algorithms frequently take form similar projected power method boumal described alternating product observations entrywise nonlinear transformation together extra onsager correction term case see amp derivation reproduces boumal algorithm except projection onto unit circle replaced soft projection function unit disk see figure magnitude maintaining quantitative measure confidence integrating usual amp blueprint representation theory compact groups obtain broad generalization method synchronization problems multiple frequencies noncommutative groups full generality nonlinear transformation simple interpretation representation theory exponential function one drawback approach although allow general observation model insist noise pairwise measurement independent fails capture certain synchronization models multireference alignment noise group element rather pair instance noise occurs image rather independently pairwise comparison adapting amp general models left future work paper organized follows begin section outline methods simplified cases synchronization motivating approach detailed discussion prior work shortfalls section provide general algorithm general problem model designed several experiments gaussian model models presented section demonstrating strong empirical performance offer two separate derivations amp algorithm section derive algorithm simplification belief propagation section give alternative derivation nonlinear update step use provide analysis amp based standard assumptions statistical physics particular derive state evolution equations govern behavior amp use identify threshold amp achieves reconstruction namely see amp threshold pca requiring snr exceed least one frequency amp achieves better recovery error threshold section argue correctness analysis providing numerical mathematical evidence known inefficient estimators beat threshold conjecture efficient algorithm able break barrier thus concluding section exploration gaps expect exist synchronization problems driven ideas statistical physics intuition iterative methods synchronization begin discussion synchronization methods cyclic group group unit complex numbers rotations examples suffice establish intuition describe much novelty approach avoiding conceptual notational complication representation theory present general case sections discuss prior work problems depth sections develop special case algorithm synchronization problem gaussian synchronization estimate uniformly drawn signal given matrix symmetric matrix whose entries distributed independently symmetry parameter scaling signal noise comparable size spectral norm hope recover exactly hope produce estimate correlated nontrivially exists depending probability hope estimate sign thus aim achieve large value review three algorithmic methods problem spectral methods scaling spectral norm signal noise taking top eigenvector may estimated significant correlation provided large enough constant specifically generative model special case spiked wigner model eigenvalues eigenvectors spiked models among main objects study random matrix theory unit norm top eigenvector vmax correlates nontrivially specifically vmax probability squared correlation tends zero fact known true estimators reflecting sharp statistical phase transition note top eigenvector may computed power iteration follows initial guess drawn randomly iteratively compute rescaling result appropriate thus entry computed yuw imagine entry sends message yuw entry vote entry identity entry entry sums incoming votes determine new value result sign reflecting weighted majority opinion whether entry ultimately also magnitude reflecting confidence serving weight next iteration thus envision spectral method basic algorithm approach effective quantified would seem suffer two drawbacks spectral method thus exploit entrywise structure signal vertex weights grow without bound potentially causing vertices exert undue influence indeed drawbacks cause major issues stochastic block model variant model gaussian observations replaced bernoulli observations usually envisioned adjacency matrix random graph sporadically vertices dominate spectral method causing asymptotically significant losses statistical power approach projected power iteration next toward amp projected power method studied variant power iteration exploits structure iteration takes form sgn sign function sgn applies entrywise thus iteration majority vote weighted magnitudes entries weights become unbalanced iterations algorithm also way plausibly exploits structure entries empirically algorithm obtains better correlation truth average approximately see figure however noisy models method appears weaker spectral method natural explanation weakness projected power method forgets distinction vote vote thus weak entries particularly problematic low ratios votes common fact heuristic analysis similar section suggests method achieve correct threshold failing produce nontrivial correlation whenever power iteration natural next step consider iterative algorithms form applies function entrywise abuse notation also denote entrywise function instead identity function spectral method sign function projected power method might imagine continuous function performs best retaining sense confidence vote without allowing resulting weights grow without bound natural ask optimal function purpose whether resulting weights precise meaning given restriction interval one imagine treating entry sign confidence precise way expectation distribution iteration entry might obtain messages yuw others compute posterior distribution summarized expectation one compute corresponds choice transformation tanh parameter see figure figure soft threshold functions used amp solid line tanh used synchronization dashed line modified bessel functions first kind used synchronization one frequency belief propagation approximate message passing algorithm may remind reader belief propagation due context inference cavity method context statistical physics may envision problem estimating probabilistic inference graphical model vertices model represent unknown entries every pair vertices participates edge interaction based matrix entry yuw ywu specifically may computed posterior distribution observing given exp precisely factorization property graphical model captures given model belief propagation proceeds fashion reminiscent previous algorithm vertex sends message neighbor encoding posterior distribution based previous distribution direct interaction vertex consolidates incoming messages new posterior distribution given messages computed messages independent however belief propagation introduces correction approach rather letting information passed propagate back next iteration belief propagation designed pass information along paths immediately backtrack specifically iteration message based synthesis messages vertices except previous iteration algorithm differs iterative methods presented behavior fact transformation distribution message necessarily linear multiplication yuw differences reduced passing framework approximate message passing simplifies belief propagation dense models weak interactions following two observations inspired theory spin glasses interaction small scaling may pass expansion small yuw computing message mean example find mean yuw message rademacher mean distribution previous iteration linear expansion ensures main step expressed product rather explicitly computing messages computationally involved may propagate backtracking messages subtract bias due simplification concentrates well correction term called onsager correction vertex passes messages neighbors based belief iteration neighbors send return messages based new beliefs time updating belief vertex time one explicitly subtract reflected influence previous belief time turns correction necessary error contributions following simplifications one arrive approximate message passing amp algorithm algorithm amp synchronization initialize small close zero random values iterate set sum incoming messages set tanh vertex new estimated posterior mean return approximate map estimate sgn proper estimate desired denotes average squared entries detailed derivations algorithm appear sections much higher generality setting synchronization algorithm equivalent approach appears statistical optimality property proven amp state nontrivial correlation truth converges estimate achieves minimum error mmse asymptotically requirement technical likely removable amp initialized small randomness trivial correlation truth early iterations resemble pca produce nontrivial correlation log iterations statistical strength amp confirmed empirically appears produce better estimate either pca projected power method every see figure amp gaussian synchronization one frequency first step toward higher generality consider following gaussian synchronization model unit complex numbers goal estimate uniformly drawn signal given matrix hermitian matrix whose entries distributed independently complex normal distribution given parameter invariant global phase shift hope estimate ambiguity would like estimator maximizes inner product conjugated second variable many previously discussed iterative techniques adapt new case spectral methods analysis spectral method holds case thus top eigenvector achieves nontrivial correlation spectral method fails nontrivial estimation provably impossible projected power method product project entrywise onto unit circle preserving phase entry setting magnitude algorithm analyzed setting shown converge maximum likelihood estimator power method one might imagine applying entrywise function vector product preserves phase entry mapping magnitude thus vector entries live unit disk convex hull unit circle might envisioned estimates posterior expectation belief propagation amp belief propagation somewhat problematic setting messages express distribution priori clear expressed finite space however simplifications approximate message passing linearity messagepassing stage enables small summary distribution suffice need store expectation distribution single value unit disk approximate message passing takes following form algorithm amp synchronization one frequency initialize small random values unit disk conv iterate set sum incoming messages set vertex new estimated posterior mean applies function magnitude leaving phase unchanged return approximate map estimate phase proper estimate desired denotes modified bessel functions first kind function depicted figure detailed derivations algorithm appear sections much higher generality amp gaussian synchronization multiple frequencies consider following elaborate synchronization problem goal estimate spike observations independent hermitian matrices whose entries distributed independently symmetry parameters denotes entrywise kth power thus given independent noisy observations imagine tions targeting different frequencies fourier modes given two independent draws previous section spectral method applied average produce nontrivial estimate soon however multiple frequencies model nontrivial estimation provably impossible present evidence section true statistical threshold fact remain thus multiple frequencies model would seem confound attempts exploit multiple observations together however discuss amp enables obtain much better estimate possible one matrix alone let return issue belief propagation represent distributions one crude approach might discretize express density finite subset points however somewhat messy discretization preserved rotation becomes worse elaborate groups one even find arbitrarily fine discretizations group acts transitively instead could exploit rich structure fourier theory express distribution fourier series thus belief distribution vertex time express computing distributional message obtain take order probability approximation asymptotically accurate thus suffices represent distributions coefficients conjugate symmetry coefficients suffice sufficiency finite description belief distribution key insight approach crucial observation concerns remaining step consolidating incoming messages new belief distribution incoming message small perturbation uniform distribution approximation log allows express message distribution log dense subset distributions satisfies appropriate continuity assumptions discuss densities respect uniform measure fourier series address analytic technicalities add obtain new belief distribution normalization const log thus obtain fourier coefficients new belief fourier coefficients density old belief products remarkably tells correct nonlinear transformation apply iteration transformation fourier coefficients density section see alternative interpretation nonlinear transformation mmse estimator constraints valid conjugate symmetry fourier coefficients thus form entire linear space contrast densities subject constraints form nontrivial convex body latter space analogue unit disk interval preceding examples transformation fourier series function exponential together normalization forms analogue preceding functions thus arrive amp algorithm problem algorithm amp synchronization multiple frequencies vertex initialize small random values iterate set vector kth fourier components estimated posterior onsager correction compute vector kth fourier components estimated posterior densities return rounding proper estimate desired even entire posteriors represented detailed derivation found sections worth emphasizing expansion log const accurate expansion estimated vertex posteriors still allows density spiked concentrated function without suffering effects gibbs phenomenon contrast finitely many coefficients algorithm computes suffice express fourier expansion density truncated expansion based computed coefficients might even become negative conclude section noting nothing derivation depended crucially gaussian observation model choice model tells propagate beliefs along edge according product could carry larger class graphical models essential properties model enables approach adapt model expressed graphical model pairwise interactions luw interaction graph dense pair potentials individually weak pair potentials luw depend group ratio core property synchronization problem pair potentials luw function assumption approximation allows algorithm track finitely many fourier coefficients formulation amp general models form discussed next section amp general compact groups approach discussed multiple frequencies readily generalizes setting arbitrary compact group fourier theory generalized representation theory fourier characters precisely irreducible representations represent distributions expansion terms irreducible representations described theorem assumption approximation pairwise observations suffice store finite number coefficients expansion note finite groups finite number irreducible representations requirement poses restriction case geometric view follows belief propagation ideally sends messages space distributions form formal convex hull illustrated case sending messages valued infinite however space thus intractable could instead ask whether convex hull taken embedding sufficient domain messages key approach observation observations suffices take embedding described sum irreducible representations section devoted presenting amp algorithm full generality along synchronization model applies particular algorithm run general graphical model formulation section analyze performance restrict gaussian observation model section representation theory preliminaries haar measure crucial property compact groups existence normalized haar measure positive measure group invariant left right translation group element normalized measure amounts concept uniform distribution group specializes ordinary uniform distribution finite group throughout paper integrals form understood taken respect haar measure decomposition fix compact group working density functions distributions order succinctly describe use analogue fourier series decomposition theorem asserts space complex scalar functions closure span coefficients following basis furthermore orthonormal respect hermitian inner product indexed complex irreducible representations dim representations assumed unitary without loss generality inner product taken second input since want algorithm able store description function using finite space fix finite list irreducible representations use decompositions assumed use representations describe functions form exclude trivial representation list need describe functions additive constant given function often write expansion form complex matrices ranges irreducibles assume functions working expanded terms representations matrix inner product used defined coefficients function extracted integration appropriate basis functions analogy fourier theory sometimes refer coefficients fourier coefficients refer irreducible representations frequencies representations real complex quaternionic type every irreducible complex representation compact group one three types real type complex type quaternionic type need deal slightly differently particular type interested properties coefficients correspond represented function complex representation real type defined reals isomorphic realvalued representation thus assume without loss generality working case clear function integrating real conversely real term expansion real representation complex type isomorphic conjugate representation irreducible representation defined assume representations list come pairs list see integrating conversely holds real finally representation quaternionic type defined quaternions following sense even comprised blocks encodes quaternion following relation note relation respects quaternion addition multiplication furthermore quaternion conjugation negate corresponds matrix conjugate transpose matrix comprised blocks form call let quaternionic type assume without loss generality takes form let real function integrating see must also conversely real see write note note trace quaternion block real graphical model formulation section take standpoint probabilistic inference graphical model thus consider task estimating observations induce posterior probability factoring pairwise likelihoods luv ranges undirected edges without assume pair interactions luw fact function depending relative orientation group elements without loss generality take luu factorization property amounts graphical model entry corresponding vertex pair interaction luw represented edge model taking decomposition luw function allows write yuw luw exp runs irreducible representations require coefficients yuv expansion also require symmetry luv lvu means yuv yvu let matrix blocks yuv input synchronization problem simply coefficients define posterior distribution latent vector group elements goal approximately recover global group element suppose observations except finite set irreducible representations allow reduce decompositions finite amount relevant information always exclude trivial representation representation contribute constant factor pair likelihood disappears normalization without loss generality assume coefficient trivial representation always zero one might also formulate version model allows node potentials luv expressing nontrivial prior observation group element although model compatible methods long node potentials also suppress generality sake readability many synchronization problems instance sensor localization noise pairwise measurement fit graphical model formulation well synchronization problems instance based measurements independent randomness one derive pairwise information comparing measurements pairwise measurements independent noise strictly fit model described note prior work often run issue achieved strong results nonetheless indeed model closely related games model application discussed model suggests minimizing objective form without interpreting posterior likelihood defer close examination algorithms synchronization problems noise future work amp algorithm state amp algorithm algorithm takes input coefficients finite list irreducibles must contain trivial representation also representation list must also appear list algorithm state time comprised fourier coefficients updated follows algorithm amp synchronization compact groups vertex initialize small random values iterate set yuw fourier coefficients estimated posterior onsager correction denotes average entries set section nonlinear transformation exp exp takes fourier coefficients function returns exp integral fourier coefficients estimated posterior densities truncated contribution irreducibles suffice next iteration return posteriors represented rounding map estimate algorithm follows intuition section derivations found sections note iteration runs time linear input matrices runtime due products rest iteration takes time expect log iterations suffice resulting algorithm respect matrix inputs applications may derive observations pairwise compared produce edge observations hacked framework abuse probability algorithm takes time respect vertex observations however applications produce matrices factorization product performed time gaussian observation model amp algorithm handles general graphical model formulation able analyze performance detail restricted following concrete gaussian observation model first introduced generalizing gaussian models section first latent group elements drawn independently uniformly haar measure representation observe matrix matrix formed vertically stacking matrices vertices parameter frequency noise gaussian random matrix drawn goe gaussian orthogonal ensemble gue gse depending whether real type complex type quaternionic type respectively case normalized entry expected concrete real case entries complex case real imaginary parts entry quaternionic case block encodes quaternion value usual way see section noise matrices independent across representations except conjugate pair complextype representations draw randomly define note normalization signal term spectral norm noise term spectral norm special cases model appeared previously one frequency one frequency fact derives amp case proves optimal introduced general model presents statistical lower bounds appendix show gaussian observation model fits graphical model formulation deriving corresponding coefficient matrices particular show scalar multiple observed gaussian matrix representation theory common examples section discuss representation theory central examples namely connect general formalism back examples section representations irreducible representations groups described discrete fourier transform fourier series respectively frequencies indexed given complex number representations complex type say frequencies refer frequencies along conjugates frequencies similarly identify complex lth roots unity frequencies defined way except avoid redundancy restrict range follows odd allow along conjugates negations even representations along conjugates plus additional representation frequencies means take frequencies along conjugates applicable case see tanh function amp algorithm section arises special case nonlinear transformation occurring amp nontrivial representation parity representation acts context first input fourier series respect uniform measure const evaluate obtain values compute exponential point obtain density normalizing density values new parity coefficient log eparity tanh representations group one irreducible representation odd dimension thus representation trivial representation representation standard representation rotations space representations real type may described action rotations space homogeneous degree spherical harmonics frequently literature instance molecular chemistry complex basis spherical harmonics given representation matrices wigner dmatrices however representation defined reals demonstrated real orthogonal basis spherical harmonics see section detailed account cases often refer synchronization problems frequencies observations assumed first nontrivial irreducibles experimental results present brief empirical exploration statistical performance amp various settings compared algorithms figure comparison iterative algorithms gaussian synchronization horizontal axis represents ratio vertical axis depicts ground truth rounded output algorithm top bottom measured left side projected power iteration green power iteration red spectral method blue amp cyan data point average trials vertices figure compare performance spectral method projected power iteration power iteration without onsager correction full amp see sections gaussian synchronization spectral method achieves optimal threshold nontrivial recovery possible achieve optimal correlation afterwards projected power method appears asymptotically achieve optimal correlation performs worse spectral method small offers reasonable improvement full amp algorithm strictly outperforms methods reflects optimality result highlights necessity onsager term gains fairly modest setting increase complicated synchronization problems figures compare performance amp gaussian synchronization multiple frequencies see section model sharp contrast spectral methods offer reasonable way couple frequencies together amp produces estimate orders magnitude accurate possible single frequency figures see similar results gaussian model section also demonstrates amp effective synchronization algorithm complicated lie groups ability exploit multiple frequencies represents promising step toward improved algorithms microscopy may viewed synchronization problem previous approaches problem effectively observations single frequency apply spectral method experiments figures demonstrate algorithm stands compelling chance achieving reconstruction remark numerical issues arise computing nonlinear transformation amp algorithm involves integration group implementation based evaluating discretization group taking pointwise exponential thus approximating integral discrete sum approach somewhat crude appears work figure gaussian synchronization frequencies bottom top ratios equal common value given horizontal axis curve depicts correlation ground truth amp estimate data point average trials vertices figure vertical axis depicts logerror top bottom figure gaussian synchronization frequencies bottom top ratios equal common value given horizontal axis curve depicts squared correlation ground truth amp estimate matrices block encodes element via standard representation rotation matrices data point average trials vertices figure vertical axis depicts top bottom error adequately experiments undoubtedly room numerical procedure improved sophisticated methods may necessary obtain adequate results lie groups note also vertex posterior question extremely concentrated near point numerical value integral depend significantly whether spike lies near discretization point however affect numerator denominator integrals approximately equal factors minimal effect normalized value derivation amp belief propagation section derive general amp algorithm section starting belief propagation similarly begin belief propagation update step see writing messages densities respect haar measure dgw luw dgu dgw denotes timestep appropriate normalization constant expand positive probability density exponential function expressed expansion exp dgu extract fourier coefficients integrating basis functions assume trivial representation log dgu dgu log dgw dgu luw dgw dgw dgu log exp yuw dgw yuw small pass linear expansion incurring error dgw dgw dgw dgu yuw dgw dgw dgu yuw dgw progress expand middle factor integrand yuw yuw yuw returning previous derivation yuw dgu yuw yuw yuw yuw log dgu dgw dgw dgw dgw dgw dgw yuw dgw dgw matrix form let dgu dgw dgu dgw dgw dgw denote transformation nontrivial fourier coefficients fourier coefficients dgu yuw map amounts exponentiation evaluation basis except trivial fourier coefficient missing input causing unknown additive shift corresponds unknown multiplicative normalize thus amounts shift output correct noting exponentiation followed normalization explicitly let exp triv denotes trivial representation rtriv onsager correction section complete derivation amp replacing nature onsager correction term reducing number messages similar derivation appendix order remove nature amp recurrence let define yuw yuv substituting yuw ywu yuw yuw yuw denotes total derivative yuw assumption consists noise signal error incurred two steps thus reach entirely recurrence first term step second term onsager correction remains simplify focus single matrix coefficient correction onsu yuw ywu yuw cef yuw cef ywu ywu note ywu yuw relation entries consist independent noise signal hence terms contributing sum order yuw ywu dependent thus onsu similarly derivation suppose depends sufficiently little treat constant onsu interlude understanding derivatives exp particular exp note following convenient identity exp exp exp exp itriv recalling itriv itriv thus obtain following form onsager correction ons amp iteration reading yuw ons mmse derivation state evolution goal section derive state evolution equations govern behavior amp gaussian synchronization model section large limit along way give alternative derivation algorithm excluding onsager term shows nonlinear function interpretation mmse minimum mean squared error estimator derivation similar based ideas first introduced give proof state evolution equations derived correct amp obeys argue correctness section mmse estimator begin defining scalar problem simplification gaussian synchronization model attempt recover single group element noisy measurements able analyze gaussian synchronization model connection simpler model idea single letterization information theory suppose unknown group element drawn uniformly haar measure irreducible representation list given measurement constants matrix gaussian entries real complex depending type entries blocks independent entry normalized expected note block matrix section complex type get measurement one representation conjugate pair define mmse estimator minimizing matrix mean simply conditional expectation squared error exp exp kuq kuq exp huq huq exp exp hwq exp hwq ranges irreducible representations list includes representations complex type likelihoods used computation derived similarly appendix recognize rescaling function amp update step amp update step consider gaussian observation model section similarly update step without onsager term indicates timestep defined based state evolution amp state block vertex applied blocks separately motivate amp update step notice similarity mmse estimator state evolution idea state evolution amp iterates approximately modeled signal plus noise namely postulate constants gaussian noise matrix block independently distributed like scalar model conjugate pairs recall blocks ground truth note sheds light amp update step iteration given uqt noisy copy ground truth first thing apply mmse estimator entrywise derive recurrence parameters change one iteration assume noise independent timestep assumption far true however turns amp onsager term corrects words derive state evolution omitting onsager term assuming independent noise timestep run amp onsager term noise timestep behaves according state evolution derive state evolution first focus signal term drawn haar measure gaussian matrix appropriate type section define second matrix expression see shortly multiple identity write signal term therefore new signal parameter take short detour state properties prove appendix lemma real multiple identity furthermore following equivalent formulas iii ezq idq ezq denotes argument previous line returning state evolution focus noise term entry matrix gaussian variance expected entry approximately therefore new noise parameter summarize state evolution recurrence simplified amp update step note state evolution recurrence implies relation provided initial values satisfy relation always arranged scaling initial appropriately amp update step without onsager term becomes uqt convenient implement amp without keeping track state evolution parameters also note variant amp matches original derivation rescaling excluding onsager term reduction single parameter per frequency rewrite state evolution recurrence terms single parameter per frequency parameter introduced earlier recall state evolution recurrence therefore update step using part iii lemma write ezq idq final form state evolution recurrence relation summarized expect state evolution recurrence exactly governs behavior amp large limit although derivation heuristic discuss correctness section caveat regarding initialized see section practice imagine initial value small random vector note initialization problematic state evolution never leave zero expect state evolution converges fixed point recurrence complications arise multiple fixed points see section expect unique fixed point reached small initialization fixed point describes output amp sense following postulate state evolution final amp iterate approximately distributed terms scaling see precise sense expect true note one use translate value measure performance mse gives exact asymptotic characterization performance amp set values prominent feature amp performance threshold derive next section one check state evolution recurrence matches cavity replica predictions one frequency indeed expect amp statistically optimal settings many others see section proven rigorously threshold section use state evolution occurrence derive threshold amp achieves nontrivial recovery particular frequencies amp fixed point equal zero vector amp gives trivial performance random guessing large limit hand least one frequency nonzero amp achieves nontrivial recovery zero vector always fixed point state evolution whether amp achieves nontrivial performance depends whether zero vector stable unstable fixed point therefore consider regime small input small approximate hwq hwq wqcd qcd wqcd qcd wqcd qcd qcd means state evolution update step becomes ezq idq ezq means small nonzero shrinks towards zero grows magnitude conclude threshold correctness state evolution section justify heuristic derivation state evolution previous section argue correctness first discuss prior work provides rigorous foundation methods used related settings show numerically amp algorithm obeys state evolution equations rigorous work state evolution state evolution introduced along amp later proven rigorously amp obeys state evolution large limit particular formal sense certain forms amp iteration particular synchronization gaussian noise special case model falls framework thus admits rigorous analysis although proofs consider case amp stated proof extends case covers synchronization model one frequency order cover general formulation amp group number frequencies one needs replace complex numbers different real algebra namely product matrix algebras expect generalization follow existing methods however additional caveat involving initialization state evolution practice initialize amp small random values recall need recover group elements global exists favorable global random initialization correlation truth however correlation corresponds large limit means technically formal proof state evolution say tells fixed amp achieves iterations large limit instead would like show iterations achieve nonzero appears proving would require analysis amp may appear initialization issue fixed initializing amp spectral method achieves correlation truth however appear easily work due subtle issue correlation noise iterates practice initialization issue actually issue small random initialization amp consistently escapes trivial fixed point provided exceeds one way explain amp messages small nonlinear function essentially identity see section amp essentially power method roughly means amp automatically initializes output spectral method experiments state evolution present experimental evidence amp obeys state evolution equations figure show two experiments one one cases see performance amp closely matches state evolution prediction see discrepancy near threshold attributed fact running amp finite whereas state evolution describes behavior gaps various settings shown using standard methods statistical physics analysis amp state evolution yields complete picture various phase transitions occur computational problem settings certain features predictions confirmed rigorously section use methods give predictions gaps gaussian synchronization model section seen large limit amp achieves nontrivial recovery least one frequency section see sometimes statistically possible succeed threshold although known efficient algorithm achieves rigorous analysis log error log error figure amp compared state evolution equations experimentally left frequencies top bottom solid line amp dotted line state evolution prediction horizontal axis ratio take equal frequencies vertical axis natural logarithm error defined error truth rounded output amp particular log error value zero top figure indicates trivial recovery random guessing lower values better right one frequency error measured error matrices whose blocks encode elements via standard representation rotation matrices inefficient estimator indeed confirmed threshold beaten cases computations section give sharp predictions exactly possible free energy recall parameter state evolution recurrence captures amount information amp current state frequency indicating information indicating complete knowledge important quantity bethe free energy per variable also called replica symmetric potential function state gaussian synchronization model given constants log exp matrix standard gaussians appropriate type real complex quaternionic depending drawn haar measure group include derivation expression computed belief propagation replica calculation roughly speaking interpretation bethe free energy objective value amp trying minimize amp thought starting origin performing gradient descent free energy landscape reaches local minimum value minimum describes final state amp shown fixed points state evolution recurrence precisely stationary points bethe free energy standard types problems conjecture amp optimal among algorithms however restriction efficiency optimal estimator given global minimum free energy shown rigorously related problem matrix estimation intuition optimal estimator use exhaustive search enumerate fixed points amp return one lowest bethe free energy note compute value minimizes bethe free energy mean achieve efficient algorithm represents correlation amp iterates ground truth since truth unknown hard find iterates prescribed examples examine bethe free energy landscapes specific synchronization problems various values discuss implications primary examples various numbers frequencies discussed section recall references frequencies means observations fourier modes first example single frequency shown figure see problem transitions statistically impossible easy amp achieves nontrivial recovery computationally hard regime particular amp statistically optimal every value figure free energy landscape frequency left global minimum free energy occurs indicating amp estimator achieves zero correlation truth right global minimum occurs nonzero achieves amp therefore amp achieves statistically optimal mse mean squared error mse departs continuously zero threshold next example problem exhibits computational gap hard phase figure take alternating group irreducible representation rotational symmetries tetrahedron amp achieves statistically optimal performance sufficiently large amp gives trivial performance statistically optimal estimator gives nontrivial performance means computational gap values amp threshold nontrivial recovery statistically possible next move problems vector figure see example computational gap example computational gap note free energy landscape amp threshold reveals whether computational gap exists gap global minimum free energy occur origin state experimental results regarding synchronization problems computational gaps first frequencies gap iff frequencies gap gap gap frequencies gap iff gave rigorous lower bounds gaussian synchronization problems showing impossible impossible hard easy figure free energy landscape frequency standard representation rigid motions tetrahedron global minimizer estimator achieves nontrivial recovery new local minimum free energy appeared global minimum still nontrivial recovery remains impossible amp stuck inefficient statistically optimal estimator achieves nontrivial global minimum amp statistically optimal computational gap appears point global minimizer transitions discontinuously positive value amp achieves optimal recovery amp value transitions discontinuously zero optimal instance one frequency statistically impossible results predict results unable show rigorously two frequencies one frequency statistically impossible threshold examples saw every amp gives trivial performance exceeds amp gives statistically optimal performance however behavior complicated namely amp exhibit nontrivial performance figure show example frequencies figure free energy landscape problems critical value darker colors indicate lower free energy left frequencies origin global minimizer free energy computational gap nontrivial recovery statistically impossible right frequencies global minimizer marked lie origin computational gap regime nontrivial recovery statistically possible yet amp fails figure example amp gives nontrivial performance take frequencies set since visualize free energy landscape dimensions instead plot state evolution recurrence evolves time number iterations two different starting points bottom two curves correspond amp performance initialize small solid line dashed line representative top two curves correspond warm start see warm start state evolution converges different fixed point larger values thus better correlation truth furthermore fixed point lower free energy shown lower one indicating optimal estimator outperforms amp acknowledgements authors would like thank amit singer roy lederman yutong chen nicholas boumal others amit singer group yash deshpande helpful discussions references emmanuel abbe afonso bandeira annina bracher amit singer decoding binary node labels censored edge measurements phase transition efficient recovery ieee transactions network science engineering emmanuel abbe afonso bandeira georgina hall exact recovery stochastic block model ieee transactions information theory amit agrawal ramesh raskar rama chellappa range surface reconstructions gradient field european conference computer vision pages springer arash amini martin wainwright analysis semidefinite relaxations sparse principal components ieee international symposium information theory pages ieee afonso bandeira convex relaxations certain inverse problems graphs phd thesis princeton university june afonso bandeira nicolas boumal amit singer tightness maximum likelihood semidefinite relaxation angular synchronization afonso bandeira yutong chen amit singer games compact groups orientation estimation may afonso bandeira moses charikar amit singer andy zhu multireference alignment using semidefinite programming proceedings conference innovations theoretical computer science pages acm jean barbier mohamad dia nicolas macris florent krzakala thibault lesieur lenka zdeborova mutual information symmetric matrix estimation proof replica formula florent raj rao nadakuditi eigenvalues eigenvectors finite low rank perturbations large random matrices advances mathematics christopher bishop bayesian pca advances neural information processing systems pages mohsen bayati andrea montanari dynamics message passing dense graphs applications compressed sensing ieee transactions information theory nicolas boumal nonconvex phase synchronization quentin berthet philippe rigollet complexity theoretic lower bounds sparse principal component detection colt pages quentin berthet philippe rigollet optimal detection sparse principal components high dimension annals statistics theodor tammo tom dieck representations compact lie groups volume springer science business media yuxin chen emmanuel projected power method efficient algorithm joint alignment pairwise differences mihai cucuringu yaron lipman amit singer sensor network localization eigenvector synchronization euclidean group acm transactions sensor networks tosn jon cohen view hiv good true science ronald coifman yoel shkolnisky fred sigworth amit singer reference free structure determination eigenvectors center mass operators applied computational harmonic analysis yash deshpande emmanuel abbe andrea montanari asymptotic mutual information binary stochastic block model ieee international symposium information theory isit pages ieee yash deshpande andrea montanari optimal sparse pca ieee international symposium information theory pages ieee yash deshpande andrea montanari finding hidden cliques size nearly linear time foundations computational mathematics david donoho arian maleki andrea montanari algorithms compressed sensing proceedings national academy sciences david donoho arian maleki andrea montanari message passing algorithms compressed sensing motivation construction ieee information theory workshop itw pages yash deshpande andrea montanari emile richard principal component analysis advances neural information processing systems pages daniel egloff markus leippold liuren term structure variance swap rates optimal variance swap investments journal financial quantitative analysis delphine sandrine largest eigenvalue rank one deformation large wigner matrices communications mathematical physics arvind giridhar kumar distributed clock synchronization wireless networks algorithms analysis proceedings ieee conference decision control pages ieee michel goemans david williamson improved approximation algorithms maximum cut satisfiability problems using semidefinite programming journal acm jacm bruce hajek yihong jiaming achieving exact cluster recovery threshold via semidefinite programming ieee transactions information theory adel javanmard andrea montanari state evolution general approximate message passing algorithms applications spatial coupling information inference adel javanmard andrea montanari federico phase transitions semidefinite relaxations proceedings national academy sciences florent krzakala jiaming lenka mutual information matrix estimation thibault lesieur florent krzakala lenka zdeborov mmse probabilistic matrix estimation universality respect output channel annual allerton conference communication control computing allerton pages ieee thibault lesieur florent krzakala lenka phase transitions sparse pca ieee international symposium information theory isit pages ieee robert litterman jose scheinkman common factors affecting bond returns journal fixed income arian maleki laura anitori zai yang richard baraniuk asymptotic analysis complex lasso via complex approximate message passing camp ieee transactions information theory frank mcsherry spectral partitioning random graphs foundations computer science proceedings ieee symposium pages ieee marc mezard andrea montanari information physics computation oxford university press marc giorgio parisi virasoro model replica solution without replicas europhys lett andrea montanari emile richard principal component analysis message passing algorithms sharp asymptotics ieee transactions information theory andrea montanari subhabrata semidefinite programs sparse random graphs application community detection proceedings annual acm sigact symposium theory computing pages acm zongming yihong computational barriers minimax submatrix detection annals statistics judea pearl fusion propagation structuring belief networks artificial intelligence amelia perry alexander wein afonso bandeira ankur moitra optimality suboptimality pca spiked random matrices synchronization sundeep rangan alyson fletcher iterative estimation constrained matrices noise ieee international symposium information theory isit pages ieee cynthia rush ramji venkataramanan finite sample analysis approximate message passing rubinstein wolansky reconstruction optical surfaces ray data optical review amit singer angular synchronization eigenvectors semidefinite programming applied computational harmonic analysis amit singer yoel shkolnisky structure determination common lines eigenvectors semidefinite programming siam journal imaging sciences david thouless philip anderson robert palmer solution solvable model spin glass philosophical magazine expansion gaussian observation model section show gaussian observation model fits graphical model formulation deriving corresponding coefficient matrices particular show scalar multiple observed gaussian matrix write log luv log consider representation separately three cases three types representations see section convenience recall gaussian observation model restricting submatrix muv real type wuv let real type recall case entry wuv log muv muv const denotes frobenius norm additive constant last step depends muv thus coefficients yuv muv complex type consider representation complex type along conjugate recall case entry wuv independent real imaginary parts drawn const muv muv log muv therefore quaternionic type consider representation quaternionic type recall case wuv block encodes quaternion value whose entries drawn independently note following relation norm quaternion corresponding matrix muv muv const log const muv denotes real part last step used fact muv inner product real see section therefore proof lemma see equal recall interpretation conditional expectation stands related nishimori identities statistical physics following symmetry properties lemma therefore define ezq proof part straightforward computation using definition part follows part distribution return proof lemma equality iii follows part lemma equality follows part lemma combining equality shown equality iii remains show real multiple identity letting identity schur lemma means possibly complex multiple identity iii done see multiple real note trace real
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arxiv dec theory practice logic programming may distributed www programming using prolog pillow daniel cabeza manuel hermenegildo clip group http http facultad universidad madrid upm del monte madrid spain dcabeza herme abstract discuss practical point view number issues involved writing distributed internet www applications using systems describe pillow publicdomain internet www programming library systems designed order simplify process writing applications pillow provides facilities accessing documents code www parsing manipulating generating html xml structured documents data producing html forms writing form handlers processing templates important contribution pillow model code thus content www pages terms pillow library developed context ciao prolog system adapted number popular systems supporting functionality also describe use concurrency highlevel model interaction ciao prolog active modules context www programming propose solution downloading execution prolog code using generic browsers finally also provide overview related work topic keywords www html xml cgi http distributed execution constraint logic programming introduction wide diffusion internet popularity world wide web protocols effectively providing novel platform facilitates development new classes portable distributed applications good support network connectivity protocols communication architectures novel platform obviously requirements programming tool useful arena however alone may enough seems natural significant parts paper expanded improved version cabeza hermenegildo cabeza hermenegildo cabeza hermenegildo cabeza cabeza hermenegildo network applications require symbolic numeric capabilities necessarily related distribution important capabilities example symbolic information processing dealing combinatorial problems natural language processing general logic programming kowalski colmerauer constraint logic programming clp systems jaffar lassez van hentenryck colmerauer dincbas van hentenryck ecrc shown particularly successful tackling issues see example proceedings recent conferences practical applications prolog practical applications constraint technology seems natural study technology fares developing applications operate internet fact prolog concurrent constraint based extensions logic programming languages general many characteristics appear set particularly well placed making impact development practical networked applications ranging simple quite sophisticated notably systems share many characteristics recently proposed network programming tools java including dynamic memory management structure pointer manipulation robustness compilation bytecode furthermore unlike scripting application languages currently proposed shell scripts perl java etc systems offer quite unique set additional features including dynamic databases search facilities grammars sophisticated well understood semantics addition systems also already offer kind low level support remote communication using internet protocols generally involves providing sockets ports interface whereby possible make remote data connections via internet native protocol systems support higherlevel communication layers top interface including blackboards sicstus prolog carlsson ciao carro hermenegildo cabeza hermenegildo hermenegildo clip group hermenegildo hermenegildo hermenegildo bueno tarau bosschere etc shared communication chikayama akl janson haridi smolka ciao hermenegildo cabeza hermenegildo cases functionality provided via libraries building top basic primitives case example sicstus ciao distributed interfaces fact shown previous work sharedvariable based communication also implemented conventional systems via library predicates using attributed variables hermenegildo cabeza hermenegildo addition communication primitives several systems offer concurrency even abstractions distributed objects mobile code useful developing distributed applications concrete interest www applications applications generally distributed www programming using prolog pillow lib use specific protocols http ftp data formats html xml application architectures cgi interface different protocols typically used types distributed applications paper study good support protocols data formats architectures provided systems building widely available interfaces basic protocols aim discuss practical point view number new issues involved writing www applications using systems well architecture typical solutions process describe pillow programming logic languages web public domain programming library systems argue significantly simplifies process writing applications pillow provides facilities generating structured documents handling herbrand terms producing html forms writing form handlers processing templates accessing parsing www documents etc also describe architecture relatively sophisticated application classes using model clientserver interaction active modules cabeza hermenegildo finally describe architecture automatic code downloading local execution using library generic browsers apart tutorial value paper present number technical contributions include idea representing html xml code structured text general prolog terms use logical variable terms leading model html template pair comprising term free variables dictionary associating names variables notion active logic modules application solving efficiency issues cgi interaction simple way idea prolog scripts application cgis identification number features added existing systems order facilitate programming www applications mainly concurrency argument throughout paper small limitations functionality disappear concurrency added systems akl ciao prolog possible add extremely useful programming layer system without making significant changes implementation argue layer simplify generation applications systems including active www pages search tools content analyzers indexers software demonstrators collaborative work systems muds moos code distributors etc purpose paper also serve tutorial containing sufficient information developing relatively complex www applications prolog clp languages using pillow library pillow library developed context ciao prolog system adapted number popular systems supporting functionality ciao prolog system pillow library freely downloaded http http cabeza hermenegildo http server www browser std doc request output fig cgi interface writing basic applications simplest way writing www applications use common gateway interface cgi cgi executable standard executable file http server program responds http requests machine serves www site tell fact contains program run rather document text sent client browser usual file distinguished belonging special directory commonly named special filename ending normally set configuration http server basic idea behind cgi interface illustrated figure user selects address cgi executable document http perhaps http browser issues standard document request http server recognizing cgi executable rather document starts executable execution stores output executable buffer upon termination executable contents buffer format browser handle html returned browser normal page content accessed following example simple executable written language source might main write write html write hello write actual executable could generated usual example ciao note examples presented order shorten html code may slightly simplified result may completely however examples used popular browsers distributed www programming using prolog pillow lib system using standalone compiler writing unix shell ciaoc executable placed appropriate place accessible via http address browser right permissions executed server example systems means executable user nobody systems make executables saved states usually disadvantage generally large size system prompt one could create executable writing something like compile save main scripts cgi applications cgi executables often programs perform relatively simple tasks added slow speed network connection comparison executing program makes program execution speed less important made scripting languages shell scripts perl popular writing programs popularity due fact compilation necessary extensive string handling capabilities also play important role case perl thus changes updates program imply editing source file logic languages priori excellent candidates used scripting however relative complication making executables needing systems start compile consult file make saved state often large size resulting executables may deter cgi application programmers appears convenient provide means programs executable scripts even reduced performance generally relatively easy support scripts functionality systems ciao program also adapted sicstus hermenegildo accomplishes task first loading file given first argument skipping first lines avoiding loading messages starting execution argument provides list command line options example unix system following program run directly script without need compilation main write write html often convenient use options ciaoc generate standalone executable independent libraries example grammars databases greatly simplify many typical applications cabeza hermenegildo form http www browser http server form data form reply fig forms interface write hello write note unix versions either program must included listing first line replaced two exec execution prolog scripts may optimized systems example ciao first time script run also compiled bytecode saved file subsequent times script changed object code retrieved file avoiding compilation interpretation overhead form handling http far shown cgi executables produce output output function input coming request obviously limited interest cgi executables become useful combined html forms html forms html documents parts html documents include special fields text areas menus radio buttons etc allow providing input cgi executables steps involved handling input contained form illustrated figure document containing form accessed via browser mosaic netscape lynx etc browser displays input fields buttons menus etc indicated document locally allows user perform input modifying fields however input ultimately handled browser instead sent handler cgi program anywhere net whose address must given form forms generally submit button pressed input distributed www programming using prolog pillow lib provided menus text areas etc sent browser http server corresponding handler two methods sending input exist get post meantime sending browser waits response program come form new html document handler program invoked much way application except information form supplied handler different ways depending system method invocation content type information encoded predefined format relates piece information corresponding field form means keyword associated field handler identifies information corresponding field original form processes responds writing html document standard output forwarded server waiting browser handler terminates important point noted simple applications handler started terminate transaction reader referred example grobe naseer complete introduction cgi scripts html forms writing form handlers llow complication writing form handlers compared writing simple cgi applications need capture parse form data said data provided several ways depending system method used invoke form encoded escape sequences relatively easy write prolog program parse input using example definite clause grammars pillow library provides predicates simplify whole task hiding protocol behind principal predicates provided include dic translates input form either post get methods even dictionary dic pairs translates empty values indicate presence attribute atom empty values one line text areas files list lines strings rest atoms numbers using implemented using dcg parsers dic var val gets value val attribute var dictionary dic fail value found simplifies merging form producers form handlers see later useful check value text area empty filters spaces newlines linefeeds val default newval useful form partially filled also first invocation combined form see section value val empty else url returns url uniform resource locator www address cgi executable method returns method method invocation form handler get post cabeza hermenegildo example suppose want make handler implements database telephone numbers queried form including single entry field name handler might coded follows include library pillow main input input name write write html title telephone database write img write telephone database name write name name write provide phone name phone write telephone number write name write write phone write telephone number available write name write phone daniel phone manuel phone sacha code quite simple hand interspersion throughout text calls write html markup inside makes code somewhat inelegant also separation computation normally desirable would much preferable encoding html code prolog terms could manipulated easily elegant way predicate translate terms html output functionality provided pillow library presented next section handling html prolog terms since systems perform symbolic processing using herbrand terms seems natural able handle html code directly terms structures distributed www programming using prolog pillow lib need translated appropriate predicates html code need output general relationship html code prolog terms allows viewing www page herbrand term predicates provide functionality pillow accepts html term list html terms sends standard output text rendering term html format chars terms also relates list html resp xml terms list ascii characters rendering terms html format predicate reversible normalizes reverse direction later uses predicate transform html terms characters implemented via dcg parsing html term certain atoms structures represent special functionality html level html term recursively list html terms following legal html terms hello hello world html term converting html terms characters translates special structures corresponding format html applying recursively arguments strings always left unchanged html terms may contain logic variables provided instantiated term translated output allows creating documents piecemeal references documents etc following sections list meaning principal prolog structures represent special functionality html level special atoms translated rest assumed normal text passed html document general structures basically html two kinds components html elements html environments html element form name attributes name name element attributes possibly empty sequence attributes either attribute name attribute assignment value html environment form name attributes text name name environment attributes form general prolog structures represent two html constructions name atts defined infix binary operator represents html element name name attributes atts atts possibly empty list cabeza hermenegildo attributes either atom structure name example term img map ismap translated html source img map ismap note html use atoms name text term functor argument text represents html environment name name included text text example term address clip translated html source address clip name atts text term functor arguments atts text represents html environment name name attributes atts included text text example term http clip home represents html source http clip home env name atts text equivalent name atts text begin name atts translates start html environment name name attributes atts exists also begin name structure useful conjunction next structure including document output generated existing piece code name pre use otherwise discouraged end name translates end html environment name name rewrite previous example follows note use logic variable response allows injecting result call output term using unification include library pillow main input input name response name response html title telephone database img telephone database distributed www programming using prolog pillow lib response using logic variable response name response name response provide phone name phone response telephone number name phone response telephone number available name phone daniel phone manuel phone sacha html construction represented structures except comments declarations could included atoms strings pillow library provides additional specific structures simplify html creation specific structures section list special structures html pillow understands many cases using general structures native html names probably good practice using specific structures sometimes convenient also structures special functionality predicate provided allows defining new structures tables layers etc specific structures include reader referred pillow manual full listing start used beginning document translates html end used end document translates produces horizontal rule translates produces line break translates produces paragraph break translates comment comment used insert html comment translates comment declare decl used insert html declaration seldom used translates decl image addr used include image address url addr translates img element image addr atts list attributes atts ref addr text produces hypertext link addr url referenced resource text text reference translates addr text label label text labels text target destination label label translates label text heading text produces heading level text text used heading useful one wants heading level relative another heading translates environment cabeza hermenegildo itemize items produces list bulleted items items list corresponding html terms translates environment enumerate items produces list numbered items items list corresponding html terms translates environment description defs produces list defined items defs list whose elements definitions prolog sequence composed operators last element sequence definition defined terms translates environment img items produces list bulleted items using image img bullet predicate provides colored bullet preformatted text used include preformatted text text list html terms element list line resulting document translates pre environment verbatim text used include text verbatim special html characters translated quoted html equivalent term includes prolog term term represented functional notation variables output used include newline html source improve human readability entity name includes entity name name special character html rather cgi protocol requires content descriptor used cgi executables including form handlers replying translates includes page graphical logo message developed using pillow web programming library points manual library source additional structures rewrite previous example follows note example use equally suitable include library pillow main input input name response name response start title telephone database image heading telephone database response distributed www programming using prolog pillow lib end response name response name response provide phone name phone response telephone number name phone response telephone number available name phone daniel phone manuel phone sacha included specific structures creating forms included explained following section specific structures forms section explain structures represent various elements related forms addr atts specifies beginning form addr address url program handle form atts attributes form method used invoke atts present method defaults post translates form addr atts specifies beginning form without assigning address handler form handler executable producing form specifies end form translates checkbox name state specifies input type checkbox name name checkbox initially checked translates input element radio name value selected specifies input type radio name name several radio buttons interlocked must share name value value returned button button initially checked translates input element input type atts specifies input type type list attributes atts possible values type text hidden submit reset translates input element textinput name atts text specifies input text area name name text provides default text shown area atts list attributes translates textarea environment option name val options specifies simple option selector name name options list available options val initial selected option val options first item selected translates select environment cabeza hermenegildo menu name atts items specifies menu name name list attributes atts list options items elements list items marked prefix operator indicate selected translates select environment example order generate form suitable sending input previously described phone database handler one could execute following goal start title telephone database heading telephone database http click enter name clip member press return input text end course one could also simply written directly resulting html document html title telephone database telephone database form post http click enter name clip member press return input text merging form producer handler interesting practice producing html forms handlers merge operation form producer handler program idea produce generalized handler receives form input parses computes answer produces new document contains answer input well new form special case must made first invocation input would empty form generated following example merges producer handler phones notice one text field exists form form submitted simply pressing return inside text field distributed www programming using prolog pillow lib include library pillow main input input name response name response start title telephone database image heading telephone database response click enter name clip member press return input text end response name response name response phone name phone response telephone number name phone response telephone number available name phone daniel phone manuel phone sacha combination form producer handler allows producing applications give impression interactive even step involves starting running handler completion note forms contain fields displayed passed input next invocation handler allows passing state one invocation handler next one finally note testing debugging cgi scripts unfortunately straightforward could useful techniques include carefully checking permissions looking data logs server replacing predicates versions print really received etc cabeza hermenegildo templates problem previous programs layout output page easily configurable source changed modifying program something normal user even expert programmer size program large may want order address pillow provides facility reading html templates also xml templates converting term format natural manipulate html template file contains standard html code slots defined given identifier means special tag slots represent parts html code html code inserted html template read pillow slots appear free logic variables corresponding pillow terms way user define layout html editor choice taking care marking left parts given names parts filled appropriately program functionality associated parsing terms encapsulated following predicate chars terms dict parses string chars contents html template unifies terms list html terms comprised template substituting occurrences special tag name prolog variables dict instantiated dictionary substitutions list pairs following example template file called assumed hold formatting output page defining html variable called response substituted response cgi program note predicate defined ciao library reads file returns second argument contents file list character codes note also calling third argument instantiated response response effect instantiating slot contents response makes use fact one slot template normally call used locate appropriate pair include library pillow library main input input name response name response contents contents response response distributed www programming using prolog pillow lib response name response name response phone name phone response telephone number name phone response telephone number available name phone daniel phone manuel phone sacha example contents template file could html head title telephone database body img telephone database response form post click enter name clip member press return input text accessing www documents facilities presented previous sections allow generating html documents including forms handling input coming forms many applications search tools content analyzers also desirable able access documents internet access generally accomplished protocols ftp http built top systems connectivity interface required protocols easily coded source language using facilities dcg parsers present http protocol supported pillow html code library uses internal representation uniform resource locators urls able manipulate easily provides predicates translate internal representation textual form facilities provided pillow accessing www documents include following predicates url info translates url url internal structure info details various components urls make predicate fail http info gives info http url http gives url http string cabeza hermenegildo url baseinfo info translates relative url url appears html page referred baseinfo given structure complete structure info absolute urls translated previous predicate http info gives info http http info gives info http dic args translates list pairs dic form dictionary returned string args appending url pointing form handler url request response fetches document internet url uniform resource locator document given structure request list options specify parameters request response list includes parameters response request parameters available include head specify interested header timeout time time specifies maximum period time seconds wait response predicate fails timeout date get document newer date example structure represents date date tuesday january name provide field authorization scheme params provides authentication field accessing restricted sites name param functor translates field name user machine parameters returned response list include see definition information content content returns content actual document text list characters status type code phrase gives status response type informational success redirection code status code phrase textual explanation status pragma data miscellaneous data date time message sent location url document moved url server identifies server responding allow methods list methods allowed server date sender believes resource last modified distributed www programming using prolog pillow lib expires date entity considered stale type subtype params returns mime document type encoding document length length size document bytes authenticate challenges request authentication chars terms already explained predicate transforms html terms html format used way around parse html code example retrieved resulting list html terms terms normalized contains structures example simple fetch document done follows http member content note error occurs document exist moved example simply fail following call retrieves document modified since october http wednesday october last one retrieves header document timeout seconds get last modified date http head timeout member date following simple application illustrating use example defines url badlinks predicate fetches html document pointed url scours check links produce errors followed list badlinks contains bad links found stored compound terms form badlink link error link problematic link error error explanation given server url badlinks url urlinfo urlinfo response member text html response member content content response content terms terms urlinfo badlinks baseurl cabeza hermenegildo baseurl baseurl env anchoratts baseurl member anchoratts url baseurl env baseurl baseurl url baseurl url baseurl urlinfo urlinfo status phrase status success name phrase name url badlink url status phrase url head timeout response member status status phrase response timeout timeout providing code www facility easily built top primitives presented far remote www modules program modules reside net particular http address way normal program modules reside particular location local file system allows example always fetching recent version given library pillow program compiled example form handler section rewritten http main input input name would load current version library time executed generalized module declaration syntactic sugar using document distributed www programming using prolog pillow lib form http http server www browser form data form reply active module predicate argn fig forms interface using active modules fetch using followed standard declaration obviously interesting combine facility caching strategies interesting straightforward implement additional feature fetch remote generally done available possible two systems use normally checked easily bytecode also may interesting combine type code downloading www document accesses code downloaded automatically particular document fetched issue addressed section finally obvious security issues related downloading code general addressed standard techniques security signatures model interaction active modules despite power interface also shortcomings serious perhaps fact handler started expected terminate interaction two disadvantages first state preserved one query next however mentioned fixed passing state form using hidden fields saving temporary file server side using cookies etc second importantly starting stopping application may inefficient example idea query large database natural language understanding system may take long time start stop system order avoid propose alternative architecture applications similar idea although based idea active modules proposed independently ken bowen bowen basic idea illustrated figure operation identical standard form handlers illustrated figure step step handler started application rather interface actual application running continuously thus contains state thus cabeza hermenegildo interface started stopped every transaction interface simply passes form input received server running application forwards output application server terminating application continues running interface application written using predicates presented interface simple script application typically compiled interesting issue communication interface application course done sockets however cleaner much simpler alternative concept active modules cabeza hermenegildo used advantage application active module active object modularity implemented via objects ordinary module computational resources attached example process unix machine resides given socket address compiling active module produces executable running acts server number relations predicates exported module relations exported active module accessed program network simply loading module thus importing remote idea process loading active module involve transferring code rather setting things calls local module executed remote procedure calls active module possibly network except compiling special way active module identical programmer point view ordinary module also program using active module imports uses way module except uses rather see also active module address network address must known order use address announced active module started via file name server would another active module fixed address present constructs related active modules ciao module predicates declaration used import predicates list predicates active module module point code written standard declaration used declaration needs following predicate accessible module module address predicate must return address address module active module imported code number standard libraries defining versions predicate address predicate define way publish address used active modules name active module taken name current executable number standard libraries defining versions predicate correspondence libraries define versions previous predicate also possible provide active modules via www address however find straightforward simply use socket addresses case generally hidden inside access method thus made transparent user distributed www programming using prolog pillow lib modulefile publishmodule makes active module executable module residing modulefile using address publish module name publishmodule executable run example operating system level module socket created hook predicate mentioned supposed defined publishmodule called order export active module address required standard driver run attend network requests module exported predicates note code modulefile need written special way scheme flexible allowing completely configure way active modules located accomplished writing pair libraries one defining way active module address published second defining way address given active module found example ciao standard libraries include example implementation libraries uses directory accessible involved machines via nfs store addresses active modules predicate examines directory find required data solutions provided examples include posting address www address implementation name server another active module one known fixed address records addresses active modules supplies data modules import serving contact agency servers clients implementation point view active modules essentially daemons prolog executables started independent processes operating system level ciao system library communication active modules implemented using sockets thus address active module unix socket machine requests execute goals module sent socket remote programs request arrives process running active module takes executes returning socket computed results results taken remote processes thus compiler finds declaration defines imported predicates remote calls active module example predicate imported active module predicate would defined compiling following code active module writing ciao toplevel using standalone compiler executing ciaoc creates executable started process example typing unix shell prompt saves address socket file waits queries module imports module also provides predicate dynamically add information database module cabeza hermenegildo response name response name response provide phone name phone response telephone number name phone response telephone number available name name phone assert phone name phone dynamic phone daniel phone manuel phone sacha following simple script used executable active module interface previous active module started process form input issue call automatically handled active module produce new form terminating locate address active module via predicate defined library library include library pillow main input input name response name response start title telephone database image heading telephone database response click enter name clip member press return distributed www programming using prolog pillow lib input text end many enhancements simple schema brevity sketched one add concurrency active module whatever means handling interaction used order handle queries different clients concurrently easy systems support concurrency natively ciao akl feel ciao offer advantages area offers compatibility prolog clp systems time efficiently supporting concurrent execution clause goals via local distributed threads carro hermenegildo goals communicate different levels abstraction shared fact database similarly blackboard shared variables also supports threads somewhat different communication mechanisms tarau bosschere finally shown szeredi also possible exploit concurrency present prolog systems aurora implementing multitasking server also interesting set things single active module handle different forms done even dynamically capabilities active module augmented fly able handle new form designating directory code loaded active module would put active module consulting directory periodically increase functionalities finally another important issue addressed providing security ensuring allowed clients connect active module case remote code downloading standard forms authentication based codes used automatic code downloading local execution section describe architecture using facilities presented previous sections allows downloading local execution prolog code accessing www address without requiring special browser complementary approach giving www access active module sense provides code executed client machine java concretely functionality desire simply clicking www pointer transparently user remote prolog code automatically downloaded way queried via forms processing done locally allow http server server machine configured give specific example files hold prolog code example special suffix like side browser configured start helper application receiving data type cabeza hermenegildo form http www browser http form data form reply loaded formreply active module code formdata answerform formdata formreply loadcode fig automatic code downloading architecture application interface prolog engine execute www downloaded code acting active module sketch procedure see figure form used query downloaded code assume already loaded browser contains link points prolog code file clicking link produces download explained note browsers handle mime types modern browsers form code file could alternatively combined document however brevity describe case separate handler form specified local executable server file tells browser page type browser starts passes file example saving file temporal directory passing name process checks whether prolog engine currently running browser necessary starts one prolog engine configured active module call predicate active module loadcode file handler asks active module read code active module reads code compiles distributed www programming using prolog pillow lib waits active module complete compilation writes done message browser browser receives done message submit button form pressed following standard procedure forms browser starts process sending form data process gets form data translates dictionary formdata passes active module call exported predicate answerform formdata formreply active module processes request returns formreply www page term contains answer possibly new form process translates formreply raw html gives back browser dying afterwards subsequent queries active module accomplished either going back previous page using back button present many browsers answer page contains new query form using case procedure continues net effect approach simply clicking www pointer remote prolog code automatically downloaded local prolog engine queries posed via form answered locally prolog engine obvious security issues need taken care architecture standard authentication techniques used however since source code passed around comparatively easy verify dangerous predicates example perhaps access files executed note also possible download bytecode since supported current systems using similar approach related work previous general purpose work www programming using computational logic systems includes best knowledge publicly available library cabeza hermenegildo manual logicweb system loke davison pillow library also described previously cabeza library built cabeza hermenegildo using input naish forms code hermenegildo bueno experiments building www interface warren pereira program released publicly available www library systems announced among places internet newsgroup cabeza hermenegildo library since ported large number systems adapted several prolog vendors well used different programmers various institutions particular ken bowen ported library als prolog extended provide group processing forms alternative use active modules bowen present work essentially significant extension library main previous body work related interfacing cabeza hermenegildo logic programming www knowledge logicweb loke davison system loke davison aim logicweb use logic programming extend concept www pages incorporating programmable behavior state shares goals java also offers rich primitives accessing code remote pages module structuring aims logicweb different logicweb presented system implementation done tight integration mosaic browser making use special features browser contrast general purpose library meant used general computational logic systems offers wide range functionalities syntax conversion html logic terms access predicates www pages predicates handling forms generally somewhat lower level abstraction logicweb believe using pillow ideas sketched paper possible add quite interesting functionality offered logicweb standard clp systems shown examples including access passive remote code modules ftp http address programs automatic remote code access querying using standard browsers forms addition discussed active remote code functionality rather code exported recently larger body work topic presented workshop held topic logic programming internet joint international conference symposium logic programming also previous version paper presented work presented loke sterling based logicweb aims provide distributed lightweight databases www basic logicweb system believe pillow library used implement systems interesting ideas proposed therein briefly mentioned work szeredi proposes architecture similar active modules order handle form requests solution handling multiple requests performed using feel ciao threads natural modeling kind concurrency ideas proposed quite interesting eclipse bonnet thomsen aimed implementing internet agents offers functionality part similar ciao libraries including facilities similar active modules approach different however several respects eclipse library implements special http servers clients contrast pillow uses standard http servers interfaces using special purpose servers may interesting approach possibly allows greater functionality hand approach general requires either substitution standard server given machine setting special server different socket address standard one eclipse library also contains functionality related active modules although interface provided lower level finally papers describing interesting www applications presented regularly underline suitability distributed www programming using prolog pillow lib computational logic systems task believe ciao pillow library contribute making even easier develop applications future additional work topic logic programming internet found proceedings workshop sponsored compulognet research network reader referred tutorials papers presented two workshops information number applications libraries topics interfacing compilation computational logic systems java examples prolog systems interfaced java binprolog see http ciao bueno others calejo experimental prolog java compilers built academia see example jprolog http commercially see example prolog tools http approach quite attractive although results compete performance conventional prolog compilers open research whether improvements java performance improved compilation technology bridge gap commercial work topic interfacing prolog www addition done als system mentioned include amzi prolog webls system http lpa prologweb system http recent work using pillow includes web integrator davulcu webbase system integrates data various web sources allows users query web sources single webdb cabeza hermenegildo database management interface also within radioweb project partners developed collaboration group codish ben gurion university language describing www page layout style rules engine interpreting rules generate www sites dynamically adapt parameters user characteristics cederberg clip group additional applications developed pillow library accessed pillow www site see later page pointers proceedings previously mentioned workshops well information including technical reports tutorial regarding topic logic programming constraint programming internet maintained http conclusions future work discussed practical point view number issues involved writing internet www applications using systems described pillow programming library systems pillow provides facilities generating structured documents producing html forms writing form handlers processing templates accessing parsing www documents accessing code posted http addresses also described architecture application cabeza hermenegildo classes including automatic code downloading using model clientserver interaction active modules finally also described architecture automatic code downloading local execution using generic browsers believe ciao pillow library ease substantially process developing www applications using computational logic systems recently developed several extensions library example setting getting cookies sample applications make extensive use concurrency systems support overlap network requests also developed complementary library interfacing prolog virtual reality modeling language vrml addition included part ciao system pillow library provided standard standalone public domain library sicstus prolog prolog clp systems supporting functionality please contact authors consult www site http pillow page http download details online version pillow manual ciao prolog system also freely available http http acknowledgments authors would like thank lee naish mats carlsson tony beaumont ken bowen michael codish markus fromherz paul tarau andrew davison koen bosschere useful feedback previous versions document pillow code first versions ciao system library developed partial support acclaim esprit project subsequent development occurred context mcyt projects ella edipia mcyt esprit project radioweb collaboration eccosic fulbright references cailliau luotonen nielsen secret web communications acm bowen march personal communication available http bueno cabeza carro hermenegildo puebla august ciao prolog system reference manual ciao system documentation school computer science technical university madrid upm cabeza hermenegildo distributed concurrent constraint execution ciao system proc workshop parallelism implementation technologies utrecht utrecht madrid available http cabeza hermenegildo march html package systems spain available http distributed www programming using prolog pillow lib cabeza hermenegildo february html www interface publicly available posting available http cabeza hermenegildo april www programming using computational logic systems library proceedings workshop logic programming www cabeza hermenegildo june www database management interface prolog technical report school computer science technical university madrid upm facultad upm del monte cabeza hermenegildo varma september library programming using computational logic systems proceedings workshop logic programming tools internet applications available http calejo land opportunities pages proceedings first international conference practical application constraint technologies logic programming practical application company also available http carlsson february sicstus prolog user manual box spanga sweden carro hermenegildo concurrency prolog using threads shared database pages international conference logic programming mit press cambridge cederberg per clip group june flexible layout styling last language technical report clip radioweb project chikayama fujise sekita portable efficient implementation tick evan proc workshop parallel concurrent programming oregon colmerauer les gramaire metamorphose tech rept univ groupe colmerauer introduction prolog iii communications acm davulcu hasan freire juliana kifer michael ramakrishnan june layered architecture querying dynamic web content acm sigmod international conference management data url http bosschere another approach parallelizing prolog pages proceedings parallel computing elsevier north holland ecrc eclipse user guide european computer research center grobe naseer hasan july instantaneous introduction cgi scripts html forms available http hermenegildo april writing shell scripts sicstus prolog posting available http hermenegildo clip group methodological issues design ciao generic parallel concurrent constraint system pages principles practice constraint programming lncs hermenegildo bueno banda puebla december ciao compiler system experimentation workbench cabeza hermenegildo future systems proceedings ilps workshop visions future logic programming available http hermenegildo cabeza carro using attributed variables implementation concurrent parallel logic programming systems pages proc twelfth international conference logic programming mit press hermenegildo bueno cabeza carro banda puebla ciao compiler system experimentation workbench future systems pages parallelism implementation logic constraint logic programming commack usa nova science hermenegildo puebla bueno using global analysis partial specifications extensible assertion language program validation debugging pages apt marek truszczynski warren eds logic programming paradigm perspective jaffar joxan lassez constraint logic programming pages acm symposium principles programming languages acm janson haridi programming paradigms andorra kernel language pages international logic programming symposium mit press kowalski predicate logic programming language pages proceedings ifips loke davison logic programming web pages acm conference hypertext acm press available http dincbas simonis van hentenryck solving large combinatorial problems logic programming journal logic programming partners radioweb project july radioweb automatic generation web sites radio brodcasting industry project description technical annex technical report radioweb project bonnet bressan leth thomsen september towards eclipse agents internet proceedings workshop logic programming tools internet applications available http carro hermenegildo interfacing prolog vrml application constraint visualization pages practical application constraint technologies logic programming practical application company smolka november definition kernel dfki documentation series german research center artificial intelligence dfki loke davison sterling september lightweight deductive databases web proceedings workshop logic programming tools internet applications available http szeredi katalin scott rob september serving multiple html clients prolog application proceedings workshop logic programming tools internet applications available http tarau april binprolog posting available http van hentenryck constraint satisfaction logic programming mit press warren pereira efficient easily adaptable system distributed www programming using prolog pillow lib interpreting natural language queries american journal computational linguistics
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porcellio scaber algorithm psa solving constrained optimization problems yinyan shuai hongliang oct department computing hong kong polytechnic university hong kong china center robotics school automation engineering university electronic science technology china chengdu china abstract paper extend algorithm called porcellio scaber algorithm psa solve constrained optimization problems including constrained mixed nonlinear optimization problem extensive experiment results based benchmark optimization problems show psa better performance many existing methods algorithms results indicate psa promising algorithm constrained optimization introduction modern optimization algorithms may roughly classified deterministic optimization algorithms stochastic ones former theoretically sound problems efficient complicated problems example comes nonconvex optimization problems deterministic algorithms may good tool obtain globally optimal solution within reasonable time due high complexity problem meanwhile stochastic ones may strong theoretical basis efficient engineering applications become popular recent years due capability efficiently solving complex optimization problems including problems travelling salesman problem algorithms take important role stochastic algorithms optimization algorithms designed based observations animal behaviors example one well known algorithm called particle swarm optimization initially proposed kennedy eberhart inspired social foraging behavior animals flocking behavior birds widely used benchmark problems field stochastic optimization pressure vessel design optimization problem important benchmark problem structural engineering optimization problem constrained mixed nonlinear optimization problem recent years many bioinspired algorithms proposed solve problem widely used benchmark problems also include nonlinear optimization problem proposed himmelblau shuaili recently novel algorithm called porcellio scaber algorithm psa proposed zhang inspired two behaviors porcellio scaber paper extend result solve constrained optimization problems original algorithm proposed deals case without constraints provide improvements original psa make capable solving constrained optimization problems compare corresponding experiment results reported ones aforementioned benchmark problems case studies extensive experiment results show psa much better performance solving optimization problems many existing algorithms ending introductory section main contributions paper listed follows extend psa solve constrained optimization problems including constrained mixed discretecontinuous nonlinear optimization problem show psa better many existing algorithms solving constrained optimization problems extensive numerical experiments problem formulation constrained optimization problem cop considered paper presented follows minimize subject decision vector corresponding lower bound upper bound ith decision variable cost function algorithm original psa cost function generate initial position porcellio scaber environment condition position determined set weighted parameter decision based aggregation propensity explore novel environments initialize extremely large value initialize element vector arbitrary value maxs tep get position best environment condition arg minxkj xkj current time among group porcellio scaber minxkj xkj minxkj xkj end randomly chose direction detect detect best environment condition min worst environment condition max position xki porcellio scaber porcellio scaber determine difference respect position aggregate xki arg minxkj xkj determine explore move new position according end end output corresponding function value visualization minimized case problem convex many standard algorithms solve problem however case problem convex problem difficult solve algorithm design vector element random number defined follows xki min xki max xki min xki evidently original psa take constraints consideration thus directly used solve cops inequality constraint conversion subsection provide improvements original psa make capable solving cops original psa focuses solving unconstrained problem first incorporate inequality constraints cost function end penalty method used new cost function obtained follows original psa sake understanding original psa given algorithm aims solving unconstrained optimization problems following form minimize decision vector cost function minimized main formula original psa given follows xki xki arg min xkj xkj defined penalty parameter using large enough value unless inequality constraints satisfied term takes dominant role cost function hand inequality constraints satisfied thus addressing simple bounds terms simple bounds handled via two methods firstly satisfy simple bounds initial position porcellio scaber set via following formula rand section modify original psa provide improved psa solving cops denotes initial value jth variable position vector ith porcellio scaber rand denotes random number region realized using rand function matlab formula guarantees initial positions porcellio scaber satisfy simple bounds secondly positions porcellio scaber updated according replacing defined constrained optimization problem updated values position vector xki may violate simple bound constraints handle issue based modified evolution rule proposed follows xki xki arg min xkj xkj algorithm algorithm evaluation end end end return figure diagram showing design parameters pressure vessel vector element random number xki min xki max min besides projection function make updated position satisfy simple bound constraints mathematical definition arg denoting euclidean norm algorithm evaluation given algorithm algorithm psa cops cost function defined generate initial position porcellio scaber according environment condition position determined set weighted parameter decision based aggregation propensity explore novel environments set penalty parameter large enough value initialize extremely large value initialize element vector arbitrary value maxs tep get position best environment condition arg minxkj xkj current time among group porcellio scaber minxkj xkj minxkj xkj end randomly chose direction detect detect best environment condition min worst environment condition max position xki porcellio scaber porcellio scaber determine difference respect position aggregate xki arg minxkj xkj determine explore move new position according evaluated via algorithm end end output corresponding function value visualization psa cops based modifications resultant psa solving cops given algorithm following section use benchmark problems test performance psa solving cops case studies section present experiment results regarding using psa solving cops case pressure vessel problem subsection pressure vessel problem considered pressure vessel problem find set four design parameters demonstrated fig minimize total cost pressure vessel considering cost material forming welding four design parameters inner radius length cylindrical section thickness head thickness body note integer multiples continuous variables table comparisons best results pressure vessel problem psa means corresponding constraint violated algorithm algorithm pressure vessel problem round end end round end end end end end end return let pressure vessel problem formulated follows minimize subject evidently problem nonlinear cost function three linear one nonlinear inequality constraints besides two discrete two continuous design variables thus problem relatively complicated problem mixed optimization projection function slightly modified presented algorithm besides initialization initial positions porcellio scaber modified follows loor rand loor rand loor rand loor rand denotes jth variable position vector ith porcellio scaber loor loor function obtains integer part real number rand denotes random number region functions loor rand available matlab best result obtained using psa instances executions using various existing algorithms methods solving problem listed table note experiments porcellio scaber used parameter set maxs tep set random number standard deviation seen table best result obtained using psa better existing results besides difference best function value among ones table best function value obtained via using psa quite small case himmelblau nonlinear optimization problem subsection consider nonlinear optimization problem proposed himmelblau problem table comparisons best results himmelblau nonlinear optimization problem psa means corresponding constraint violated also one well known benchmark problems bioinspired algorithms problem formally described follows minimize subject decision vector problem nonlinear inequality represented two nonlinear inequality constraints example constraint replaced following two constraints thus problem also solved psa proposed paper best result obtained via using psa instances executions together result obtained algorithms methods listed table experiments porcellio scaber used parameter set maxs tep set random number standard deviation evidently best result generated psa ranked among results table results conclude psa relatively promising algorithm solving constrained optimization problems quite smalle performance difference psa best one may result usage penalty method constant penalty parameter conclusions paper algorithm psa extended solve nonlinear constrained optimization problems using penalty method case studies validated efficacy superiority resultant psa results indicated psa promising algorithm solving constraint optimization problems several issues requires investigation select best penalty parameter guarantees compliance constraints also optimality obtained solution besides enhance efficiency psa also worth investigating acknowledgement work supported national natural science foundation china numbers hong kong research grants council early career scheme number also departmental general research fund hong kong polytechnic university number references kennedy eberhart particle swarm optimization proc ieee int conf neural gandomi yang benchmark problems structural optimization computational optimization methods algorithms berlin prempain improved particle swarm optimizer mechanical design optimization problems eng vol gandomi yang alavi cuckoo search algorithm metaheuristic approach solve structural optimization problems eng vol pauline sin sheng kiong design optimization structural engineering problems using adaptive cuckoo search algorithm proc int conf control autom garg deep effectiveness constrained laplacian biogeography based optimization solving structural engineering design problems proc int conf soft comput problem himmelblau applied nonlinear programming new york zhang psa novel optimization algorithm based survival rules porcellio scaber available https yang firefly algorithm stochastic test functions design optimisation int vol lee geem new algorithm continuous engineering optimization harmony search theory practice comput methods appl mech vol sandgren nonlinear integer discrete programming mechanical design optimization mech des asme vol chow genetic algorithms nonlinear mixed optimization problems via parameter optimization engrg vol coello use penalty approach engineering optimization problems comput vol deb geneas robust optimal design technique mechanical component design dasgupta michalewicz evolutionary algorithms engineering applications berlin kannan kramer augmented lagrange multiplier based method mixed integer discrete continuous optimization applications mechanical design mecha design trans asme vol akhtar tai ray simulation model engineering design optimization eng vol tsai global optimization signomial discrete programming problems engineering design eng optmiz coello hybridizing genetic algorithm artificial immune system global optimization eng vol chou global approach nonlinear mixed discrete programming design optimization eng optmiz vol kaveh talatahari improved ant colony optimization constrained engineering design problems eng comput vol omran salman constrained optimization using codeq chaos soliton vol homaifar lai constrained optimization via genetic algorithms simulation vol gen cheng genetic algorithms engineering design wiley new york dimopoulos engineering optimization based evolutionary social metaphors comput method appl mech vol
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logical methods computer science vol submitted published jul relational parametricity control masahito hasegawa research institute mathematical sciences kyoto university kyoto japan presto japan science technology agency address hassei abstract study equational theory parigot connection style cps translation fragment observed relational parametricity target calculus induces natural notion equivalence hand unconstrained relational parametricity turns inconsistent following facts propose formulate relational parametricity constrained way might called focal parametricity dedicated gordon plotkin occasion sixtieth birthday introduction introduced parigot one representative term calculi classical natural deduction widely studied various aspects although still active research subject said reasonable understanding propositional good reduction theories cps semantics corresponding operational semantics also canonical equational theories enjoying semantic completeness last point overlooked complete axiomatizations provide deep understanding equivalences proofs also semantic structure behind syntactic presentation due parigot studied depth calculus classical natural deduction particular strong normalization results extensions inductive types central research topic importance strong normalization however seems attempts giving equational theory supported fine semantic structure situation rather frustrating since without equational semantic accounts discuss correctness impredicative encoding datatypes system acm subject classification key words phrases polymorphism parametricity continuations logical methods computer science masahito hasegawa creative commons masahito hasegawa subsystem several beautiful results relational parametricity universal properties impredicative constructions covariant gives initial algebra functor suitable sense certainly wish story work attempt identify equational theory backed certain semantic structures specifically propose relational parametricity principle sound sufficiently powerful deriving equivalences parametric cps semantics first consider semantics given fragment existential types conjunction types arrow types distinguished type choice target calculus due recent work fujita translation sends type variable arrow type universal type term sent considered natural extension streicher cps translation follows translation already gives reasonable equational theory validates standard fact consequence fibred version category continuations construction however starting point observe impredicative constructions target calculus satisfy certain universal properties terminal object follow relational parametricity impredicative constructions source quite way first might expect instance type give initial object instead plays role falsity type answer type fact elimination actually algebra doublenegation monad another major example give initial algebra gives initial algebra respect terms certain class terms focal terms mentioned particular free isomorphic short impredicative encodings get extra double negations relational parametricity consistent equational theory induced cps semantics consequence encode cartesian products though added easily also express classical disjunctions though added without changing target cps translation focal parametricity results suggest cps translation parametric target calculus gives reasonable semantic foundation equational theory sufficient obtaining various interesting results however parametricity used rather indirectly via cps translation also wish decent notion parametricity directly within figure sort parametricity principle expected recall following fact parametricity given polymorphic term covariant types instances say show sound complete respect result together syntactic analysis cps translation appear forthcoming paper fujita relational parametricity control obey naturality following diagram commutes longer true example let elimination exist naturality arbitrary maps implies inconsistency get every letting obvious map enough kill theory similar result observed classical proofs peirce law end look focus centre focal map algebra morphism monad mentioned map making naturality diagram elimination commute follows notion relational parametricity construction graph relations allowed focal maps consistent nontrivial models together definability fullness cps translation see least powerful parametricity cps target calculus mentioned thus gives powerful principle deriving equivalences terms actually conjecture two notions parametricity agree open writing article principle shall call focal parametricity natural notion parametricity sketch use focal parametricity deriving free theorems syntactically towards parametricity computational effects conceptual abstract level story closely resembles study linear parametricity recursion case linear parametricity graph relations allowed constructed linear maps linear map algebra map lifting monad claim like linear parametricity gives solution accommodating nontermination recursion polymorphic setting advocated plotkin focal parametricity provides way accommodating control features polymorphic setting short focal parametricity linear parametricity control future work would interesting challenge find unifying framework linear parametricity focal parametricity useful parametric polymorphism recursion control realistic programming languages ambitiously keen see adequate notion parametricity fairly general effectful settings possible starting points direction might include parametricity graphs approach allows deal parametricity general level including linear parametricity instance category masahito hasegawa linear continuations construction induces cps translation girard translation special cases see section discussions related issue construction paper rest paper organised follows section introduce calculi subject study section consider implications relational parametricity calculus focal parametricity introduced section followed examples section including focally initial algebras type church numerals section gives alternative characterisation focus suggests generalisation work theory parametricity general effects give concluding remarks section calculi given follows essentially follow parigot formulation flavour selinger types typing judgement stands typing context variables context names continuation variables relational parametricity control axioms equational theory standard ones note consider extensional theory last two axioms make uses mixed substitution instance means replacing occurances form recursively sequel frequently use following syntactic sugar first let type type falsity may also write using define named term fresh follows holds express elimination making use polymorphic classical features expected properties studied later parametricity assumptions target calculus tthe literature often taken target cps translation fujita observed actually suffices consider fragment negations conjunctions existential types target paper follow insight considered shorthand type replaced simplicity keep type constant syntax terms fairly standard one though conjunctions employ slightly less familiar elimination rule parallels existential types masahito hasegawa let let employ standard let let let let cps translation cps translation present cps translation considered extension introduced streicher rather translations plotkin parigot fujita introduce extra negations respect extensionality relational parametricity control soundness type soundness follows straightforward induction proposition type soundness note also hold equational soundness proposition equational soundness addition definability result proposition fullness proved providing inverse translation cps translation hold program continuation let let answer let let follows let let let let considered style transformation sense sabry follows exists generated grammar suffices take form moreover routinely show thus cps translation enjoys fullness terms definable modulo provable equality definability important relating parametricity principles source target calculi masahito hasegawa semantic explanation short explanation cps translation works intended readers category theoretic background categories continuations construction fibrations polymorphic type theories response category response object induces control category fibred response category finite products simple coproducts existential quantifiers induces fibred control category finite products simple products universal quantifiers let write response category assume weakening functor left adjoint subject condition thus hence regarded weakening functor right adjoint given used interpreting universal quantifier cps transformation essentially syntactic interpretation semantic construction cps semantics parametric target calculus parametricity target calculus target calculus seen subset via standard encoding conjunctions existential types define relational parametricity target calculus way logic parametricity system system one may directly define parametricity principle often called simulation principle existential type see example paper consider relations constructed graphs identity obtained following construction shall call admissible relations among admissible relations fundamental graph relations given term free variable define graph relation short iff given type whose free type variables included admissible relations define admissible relation follows identity relation terms type relation iff implies hence relation iff relation iff admissible relational parametricity control last case relation defined one may define admissible relations admissible hold let identity relation terms type relational parametricity asserts whose free type variables included implies consistency follows immediately parametricity proposition consequences parametricity derive gives terminal object unique inhabitant could rewritten gives final coalgebra occurs negatively instance last case isomorphism holds occur freely proofs standard papers cited see implications parametricity results target calculus refer induced cps translation parametric target calculus falsity type first example let consider falsity type since terminal unique inhabitant parametric target calculus obtain coincide streicher translation consequence following equations named terms validated thus type serves falsity type found formulation addition show algebra doublenegation monad term model initial algebra substantial example initial algebra positive see unfortunate clash types cause serious masahito hasegawa problem calculate suggests isomorphic double negation one might think contradicts standard experience parametricity isomorphism since subsumes however isomorphism regarding cps interpretation otherwise causes degeneracy truth term isomorphism still initial algebra shall consider issue later shall emphasize parametricity principle used least without certain constraint otherwise would isomorphism hence degeneracy follows every impredicative encodings recall impredicative encodings logical connectives cps translations parametric target calculus satisfy easily seen defined logical connectives source calculus obey standard universal properties parametric models short hence amount classical encodings cartesian product isomorphic possible add cartesian product types also need add coproduct types target calculus terminal object isomorphic add terminal object initial object target coproduct isomorphic coproduct follow hand possible enrich initial object without degeneracy selinger note control categories alternatively might relational parametricity control add classical disjunction types hence note serves unit classical disjunction work existential type isomorphic polymorphism note answer type considered constant specific property fact could used type everything defined polymorphically regarding thus apply polymorphism principle particular closed term type considered sent type way reasoning goes behind parametricity principle target calculus justified parametricity instance consider type parametricity means although terminal object unique closed inhabitant similarly thus see closed inhabitant however reasonings based polymorphism become much harder complicated types force polymorphism setting seems still obvious focal parametricity seen cps semantics respect target calculus relational parametricity induces reasonable equational theory however parametricity used rather indirectly via cps translation consider notion parametricity directly available within cps translating relations key formulating relational parametricity use graph relations terms considered representing functional relation without graph relations relational parametricity reduces basic lemma secondorder logical relations hand allow terms used constructing relations fact linear parametricity linear strict maps allowed used constructing graph relations choice allows weaker notion parametricity accommodate recursion naturally led look characterisation used graph relations without breaking soundness respect cps semantics parametric target calculus suppose allowed use graph relation term ensure soundness use graph relation shall consider cps translation relations instance hope sent relation types target calculus however since relation target calculus complete translation relations reduce parametricity principle parametricity target calculus masahito hasegawa fortunately way characterise translatable without performing modulo technical assumption known equalising requirement notion focus recall focus definition called focal algebra morphism following diagram commutes holds focal terms compose identity obviously focal equivalence classes focal maps form category hereafter shall call focus characterisation focal maps concise closely follows semantic considerations subtle problem weak establish focality important terms used polymorphic feature expressing involves falsity type axioms guarantee work properly sufficiently many focal maps parametricity principle restricted focal maps would useless see issue clearly shall look another classical combinator peirce law make use polymorphism abort map falso quodlibet defined without classical feature well known elimination expressible peirce law together falso quodlibet see also case level uniformity proofs let say repeatable focal map called type type free name correspond focal maps thus notions associated constructions cps target categories via via focus essentially relational parametricity control commutes discardable commutes proposition focal repeatable discardable note corresponding result setting observed characterisation algebraic values repeatable discardable expressions follow terminology reformulation allows see second diagram involves polymorphically defined needs justified additional hand first diagram problematic make use polymorphism additional axioms end add axioms thinking parametricity discardable discardable discardable equivalent asking also equivalent note discussed section satisfies conditions also shall note repeatable together additional axioms become focal alternatively could type constant assume standard axiomatization falsity type case defined without polymorphism problem disappears develop focal parametricity principle top additional axioms problem overlooked preliminary version paper wrongly assumed repeatability alone would imply focality masahito hasegawa parametricity principle given focal define graph relation iff also let identity relation terms type paper consider relations given graphs focal maps identity obtained following construction shall call focal relations given type whose free type variables included focal relations define focal relation follows relation iff implies relation iff holds focal relation focal relational parametricity asserts whose free type variables included implies thus departure standard parametricity principle condition graph relation construction allowed focal maps note restriction necessary apply parametricity polymorphic terms get naturality diagrams term allowed used graph relation construction consistency soundness consistency focal parametricity sense equational theory focal parametricity trivial follows fact parametric models object continuation monad satisfies equalising requirement component unit equaliser employ syntax cps target calculus internal language models cps translation considered give semantic interpretation models focal term exists unique using fact given focal relation construct admissible relation follows graph relation let hgi unique map given defined straightforward induction parameter relations replaced theorem model given focal relation implies theorem consistency focal parametricity consistent know term model parametric target calculus satisfies equalising requirement definability result parametricity target focal parametricity agree alternatively consider refined target calculus construct ensuring equalising requirement detailed taylor work sober space lambda calculus sobriety know one direction true thanks definability theorem equality derivable also derivable focal parametricity relational parametricity control examples show certain impredicative encodings satisfy universal properties respect focus using focal parametricity principle focal decomposition start remark following focal decomposition analogous linear decomposition bijective correspondence terms focal terms natural focal focal focal proposition holds focal falsity focally initial object shall proceed reason impredicative encodings first example falsity first note focal parametricity says since extensionality get suppose focal parametricity know hence hgi thus hence conclude unique focal map means initial focus focally initial algebra fairly standard encoding foldx following diagram commutes fold fold therefore weak initial however noted initial fact even isomorphism applying focal decomposition masahito hasegawa obtain commutative diagram fold fold focal show fold unique focal map making diagram commute thus initial focus sketch proof fairly analogous corresponding result parametric given first parametricity obtain implies folda fold whenever focal also corollary thanks extensionality combining observations desired result focal satisfies extensionality conclude also implies isomorphism inverse given special case letting constant functor obtain isomorphisms free calculation see inverse see isomorphism section type church numerals conclude section remark type church numerals recall parametricity initial algebra natural numbers object whose closed inhabitants equal church numerals given usual longer true observed parigot closed inhabitants equal church numerals contrast focal parametricity focally initial algebra shown way case focally relational parametricity control initial algebras spelling focal map focal exists unique focal making following diagram commute see useful observe following bijective correspondence variant focal decomposition given focal conversely focal map follows hold focal define folda follows diagram commutes focal parametricity implies folda unique focal map general characterisation far concentrated relational parametricity one may feel story specific case continuations immediately applicable computational effects section describe alternative characterisation focus makes sense extension namely show equipped monad type equipped algebra structure see case focal parametricity monad isomorphic continuation monad focal maps precisely algebra maps monad suggests natural generalisation work theory parametricity general computational effects monad let free define proposition term model forms monad masahito hasegawa one might think trivial isomorphic assume standard parametricity always case however already seen focally parametric proposition algebra monad thus canonically equipped algebra structure one may think trivial standard parametricity isomorphism inverse however case define notion linear maps terms monad canonical algebras close axiomatic domain theory characterising strict maps also control categories characterising focal maps definition linear algebra morphism holds linear holds may write linear standard parametricity every linear focal parametricity linear maps precisely focal maps see passing note following interesting observation proposition following conditions equivalent algebras determined pointwise manner linear linear note close additional axioms discussed section also note satisfies one conditions agree every linear decomposition correspondence maps linear maps focal maps algebra maps shall consider monad focal parametricity proposition focally parametric algebra monad corollary focal algebra map proposition monad isomorphic double negation monad focally parametric relational parametricity control proposition following diagram commutes focally parametric corollary linear focal thus focal map focally parametric characterised terms monad defined arbitrary believe monad deserves much attention considered trivial know characterise essential notion focus case relational parametricity presence control feature fact story end presence recursion behaves like lifting monad indeed lifting theory linear parametricity last isomorphism follows fact gives initial algebra observations suggest exists general framework similar axiomatic synthetic domain theory lifting monad replaced strong monad continuation monad example theory parametricity general computational effects built recently alex simpson made progress direction developing polymorphic type theory interpreting types algebras monad constructive universe work fits well case linear parametricity recursion plausible also explains case focal parametricity control conclusion future work studied relational parametricity first considering cps translation parametric fragment directly giving constrained parametricity later call focal parametricity seems natural parametricity principle presence controls sense linear parametricity works presence recursion remain many things addressed future previous section already discussed research direction towards relational parametricity general effects shall briefly mention future work closely related main development paper firstly yet complete precise comparison focal parametricity parametricity cps target calculus involves subtle interaction parametricity technical condition equalising requirement secondly study focal parametricity extensions observed focal parametricity many popular datatypes cartesian products classical disjunction types however added problem adding general initial algebras problematic initial object already means inconsistency might safe add certain carefully chosen instances hand final coalgebras seem less problematic though generic account still missing masahito hasegawa perhaps also need consider cps translation datatypes systematic way interesting topic discussed paper duality calculi control primitives fact straightforward consider extension short universal quantifiers studied paper amount existential quantifiers sure calculus existential quantifiers interest however good starting point understand parametric polymorphism possibly computational effects syntactic semantic aspects particular provide new insights famous difficulty accommodating continuations type system finally also consider better ideally semantic formulation focal relations paper consider coming focal maps seems natural regard subalgebra monad focal relation assume presence cartesian product looks closely related pitts relations recursion acknowledgement thank fujita discussions cooperations related work also grateful ryu hasegawa alex simpson comments discussions encouragements references abadi cardelli curien formal parametric polymorphism theoret comput sci ariola herbelin minimal classical logic control operators proc automata languages programming springer lecture notes comput sci barthe uustalu cps translating inductive coinductive types proc partial evaluation program manipulation bierman pitts russo operational properties lily polymorphic linear lambda calculus recursion proc higher order operational techniques semantics electronic notes theoretical computer science dunphy parametricity notion uniformity reflexive graphs phd thesis university illinois filinski declarative continuations investigation duality programming language semantics proc category theory computer science springer lecture notes comput sci fujita galois embedding polymorphic types existential types proc typed lambda calculi applications springer lecture notes comput sci varieties effects proc foundations software science computation structures springer lecture notes comput sci girard fonctionnelle des coupures ordre etat paris vii girard linear logic theoret comp sci harper duba macqueen typing continuations funct programming hasegawa semantics linear proc functional logic programming springer lecture notes comput sci relational parametricity control hasegawa relational parametricity control extended abstract proc logic computer science hasegawa kakutani axioms recursion symbolic comput hasegawa categorical data types parametric polymorphism math struct comp sci hofmann streicher completeness continuation models inf comput jacobs categorical logic type theory elsevier kakutani duality recursion iteration proc computer science logic springer lecture notes comput sci kakutani hasegawa parameterizations operators control categories fundam inform lambek scott introduction categorical logic cambridge university press matthes parigot second order inductive types proc typed lambda calculi applications springer lecture notes comput sci moggi notions computation monads inf comput nakazawa tatsuta strong normalization proof second order classical natural deduction symb log corrigendum symb log ong semantic view classical proofs categorical denotational characterizations preliminary extended abstract proc logic computer science ong stewart foundation functional computation control proc principles programming languages parigot algorithmic interpretation classical natural deduction proc logic programming automated reasoning springer lecture notes comput sci parigot proofs strong normalisation second order classical natural deduction symb log pitts parametric polymorphism operational equivalence math struct comp sci plotkin theoret comput sci plotkin type theory recursion extended abstract proc logic computer science plotkin abadi logic parametric polymorphism proc typed lambda calculi applications springer lecture notes comput sci power robinson premonoidal categories notions computation math struct comp sci reynolds towards theory type structure proc colloque sur programmation springer lecture notes comput sci reynolds types abstraction parametric polymorphism proc ifip world computer congress information processing sabry note axiomatizing semantics control operators tech university oregon selinger control categories duality categorical semantics calculus math struct comp sci selinger remarks control categories manuscript simpson relational parametricity computational effects manuscript streicher reus classical logic continuation semantics abstract machines funct program takeuti axiomatic system parametricity fundam inform taylor sober spaces continuations theory applications categories masahito hasegawa thielecke categorical structure continuation passing style phd thesis university edinburgh thielecke answer type polymorphism continuation passing proc european symposium programming springer lecture notes comput sci wadler theorems free proc functional programming languages computer architecture work licensed creative commons license view copy license visit http send letter creative commons nathan abbott way stanford california usa
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analysis influence network topologies local global dynamics metapopulation systems daniela besozzia paolo cazzanigab dario pescinib giancarlo maurib degli studi milano dipartimento informatica comunicazione via comelico milano italy besozzi degli studi dipartimento informatica sistemistica comunicazione viale sarca milano italy metapopulations models ecological systems describing interactions behavior populations live fragmented habitats paper present model metapopulations based multivolume simulation algorithm stochastic class membrane systems utilize investigate influence different habitat topologies local global dynamics metapopulations particular focus analysis migration rate individuals among adjacent patches capability colonizing empty patches habitat compare simulation results obtained habitat topology conclude paper proposals research issues concerning metapopulations introduction field metapopulations ecology deals study spatial systems describing behavior interacting populations live fragmented habitats purpose models understand local global dynamics metapopulation systems usually balanced local extinctions new colonizations unoccupied patches depend spatial arrangement habitat consequently relevant insights related fields ecological research evolutionary ecology conservation landscape management achieved indeed topology fragmented habitats potentially holds relevant implications persistence populations robustness natural anthropogenic disturbance recently addition ever increasing applications methods analysis complex networks cell biology graph theory also applied study metapopulations systems graph models metapopulations nodes used represent habitat patches graph edges used denote functional connections patches typically related dispersal individuals attributes associated nodes describing quality dimension patches different types edges exploited represent distance connected patches rate dispersal couple patches simply whether two patches connected metapopulation models using methods simple implement require relatively data definition models implement detailed aspects milazzo eds applications membrane computing concurrency modelling population biology eptcs besozzi influence network topologies metapopulations dynamics concerning nature interaction populations types modeling approaches useful analysis specific features metapopulations first focuses properties habitat topology second concerned emergent dynamics paper present stochastic multivolume model metapopulations integrates explicit representation interactions individuals populations therefore allows simulate emergent local global dynamics graph description habitat topology allows investigate influence distinct spatial structures dynamics model represents simplified extension previous metapopulation model introduced based multivolume stochastic simulation algorithm stochastic class membrane systems membrane systems systems introduced class unconventional computing devices distributed parallel nondeterministic type inspired compartmental structure functioning living cells basic model consists membrane structure multisets objects evolve according given evolution rules comprehensive overview systems many applications various research areas ranging biology linguistics computer science found distinct compartments multivolume model arranged according specified hierarchy membrane structure additional assumption topological structure volume dimensions change system evolution volume assumed satisfy standard requirements classical stochastic simulation algorithm see details inside volume two different types rules defined internal rules modify objects contained inside volume take place case metapopulation describe growth death population individuals according model preys predators communication rules used move objects adjacent volumes case metapopulation describe migration population individuals paper exploited analyze emergent dynamics metapopulation systems focus influence topology patches migration individuals capability colonize patches habitat purpose consider six different habitat topologies formally described graph structures analyze topological structure connections rate individual dispersal connected patches influence local global dynamics metapopulation particular first consider given topology fixed dispersal rate patches influence dynamics focus colonization empty patches starting dispersal predators live patches occupy peculiar positions given network topology paper structured follows section present concept metapopulations ecology describe multivolume model metapopulations focusing particular different habitat topologies section show simulation results concerning influence habitat topologies emergent dynamics metapopulations considering effects predators dispersal colonization finally section conclude paper final remarks several proposals research issues concerning metapopulations metapopulations section first provide brief introduction relevant features metapopulations concerning topology habitats emergent dynamics describe modeling approach used paper based stochastic class membrane systems used besozzi section analyze influence different network topologies dynamics metapopulations dynamical models interacting populations ecology since introduction concept metapopulations also called systems extensively applied ecology analyze behavior interacting populations purpose determining fragmented habitats influence various aspects systems local global population persistence evolution species lately topic largely employed populations species living natural fragmented landscapes metapopulation consists local populations living spatially separated habitats called patches characterized different areas quality isolation connected dispersal pool spatial place individuals population spend lifetime migration among patches systems two principal types dynamics exist one hand individuals different populations local interactions inside patch according given dynamical model system interaction preys predators hand dispersal individuals among mutually connected patches influence global behavior whole system dispersal individuals usually dependent distance patches may reduce local population growth thus increase extinction risk due also environmental demographical stochasticity hence persistence populations assumed balanced local extinctions process colonization establishment new populations empty patches several theoretical frameworks metapopulation analysis defined remarking specific properties systems either explicitly implicitly considered modeling methods see details instance referring landscape theoretical models take care spatial structure habitat local quality environment patch areas mutual connectivity isolation order capture effect habitat fragmentation species persistence fact good local conditions determine growth survival populations inside patches high patch connectivity decrease local extinction risk moreover dispersal colonization elements used account importance real landscape structures referring population interactions dynamics colonization depend cooperation migrating individuals first case called allee effect models accounting dynamics assuming whether patch occupied usually consider local dynamics faster time scale respect global dynamics also neglect dependence colonization extinction rates population sizes finally regional stochasticity account bad good years local environmental quality depends weather conditions affect sustenance resource availability influence growth survival populations recently models metapopulations started defined intuitive visual way hold representation ecological systems see references therein models nodes represent habitat patches graph edges denote functional connections patches typically related dispersal individuals addition attributes associated nodes describing quality dimension patches different types edges adopted represent distance connected patches rate dispersal couple patches simply whether two patches connected models allow make insights features habitat distribution predominant importance nodes clusters nodes respect characteristics metapopulation like dynamics influence network topologies metapopulations dynamics vulnerability disturbance persistence populations according dispersal results open promising perspective related research fields evolutionary ecology conservation biology epidemiology management design natural reserves model metapopulations focusing network topologies issues discussed section explicitly considered previous model metapopulations works metapopulation models based class membrane systems called dpp used execute qualitative stochastic simulations local global dynamics metapopulations particular introduced model metapopulations dynamics additional features used order catch better describe relevant properties modeled system instance regions membrane structure represented nodes weighted graph attributes weight associated edges corresponds distance among connected regions attributes specify surface dimension new features necessary order outline spatial distribution patches relevant additional features associated dimension patch needed define density populations living inside patch distance needed identify isolated patches well define dispersal rates migrating individuals moreover using rules modify objects act mute rules modified classical view maximal parallelism allowing maximal application rules time reducing maximal consumption objects model applied investigate emergent metapopulation behaviors influence patch dimension distance stochastic breeding dynamics underlying migration colonization effects due isolated patches etc extended analysis model focusing periodic resource feeding strategies compared different systems either increasing decreasing stationary purely feeding stochastic phases defined inside patch shown instance seasonal variance transform basic dynamics inside patch complex dynamics different phases feeding cycle identified effect standard oscillations preys predators section present simplified model metapopulations exploits multivolume stochastic simulation algorithm respect previous model need use concept mute rules probabilistic choice applications rules already embedded tau leaping algorithm based moreover consider presence dispersal pool instead focus analysis direct communication individuals among interconnected patches according fixed network topologies order compare influence network decided perform analysis total patches spatially arranged different ways namely assume network topologies described graphs number nodes distinct connections chain grid star ring complete random structure see graphs respectively fig refer formal data structure using term graph use term network denote topological relationship graph formally network topology generally described weighted undirected graph set nodes node characterized value set attributes kind set undirected edges nodes besozzi figure network topologies weight function associating cost edge case metapopulations set nodes coincides set patches attribute node represents area patch edges characterize patches directly reachable patch might exist well considered work weight edge represents cost measure effort individuals face moving patch given network topology denote set nodes directly connected node also denote deg degree patch number patches directly connected formally deg card outline follows assume edges cost patches dimension rational behind paper focus attention influence different topologies habitat network local global dynamics metapopulations regardless local features patch distances patches features might naturally added works related model real data used define specific model metapopulation systems addition chosen network topology model metapopulations also considers presence species individuals locally interact according chosen dynamics give rise global dynamics thanks dispersal processes purpose paper assume patch characterized model describing interaction individuals two populations namely preys predators inside patch model described following set internal rules denotes preys denotes predators denotes sustenance resources influence network topologies metapopulations dynamics empty symbol rules model growth preys predators respectively rule models death predators rule also characterized stochastic constants expressed used together current amounts individuals occurring patch evaluate application probability step step according tau leaping algorithm see details simulations shown hereafter executed using following values stochastic constants initial amount preys predators sustenance resources value fixed entire duration simulation simulations performed software biosimware implements different stochastic simulation algorithms single multivolume systems software available free download http fig show oscillating dynamics left side preys predators single patch obtained choice parameters corresponding phase space right side figures considered reference compare discuss dynamics obtained model described section individuals time figure dynamics single patch oscillations preys predators left side corresponding phase space right side single patch model extended model inside patch network topology add many communication rules number patches connected total deg rules inside patch rules needed move population individuals among various patches network thus allowing analyze effects migration colonization metapopulation done attaching destination target communication rule specifying destination patch usually done systems formally patch network add dispersal rules target similarly local rules probability applying dispersal rule determined using stochastic constant whose values given next section consider different migration rates besozzi influence network topologies metapopulation dynamics section analyze topological structure connections rate individual dispersal connected patches influence local global dynamics metapopulation particular section consider given topology fixed dispersal rate influence dynamics section focus capability colonization empty patches starting dispersal predators living patches occupy peculiar positions given network topology network topologies migration section analyze role migration compare six network topologies respect four different conditions dispersal rules namely assume patch topology initialized complete model given section value stochastic constant dispersal predators patch assume one following values deg considering first condition reference power dispersal second third condition first one irrespective position patch occupies considered network terms flux dispersal patch first three conditions results amplified number connections patch respect patches network contrary fourth condition corresponds situation patch sum values constants dispersal rules always equal rate dispersal along edge depends degree instance network topology fig value patches equal value patches equal network topology fig value patch equal value patches equal weigh dispersal predators according position patch network simulate situation flux dispersal patch towards adjacent patches uniform throughout whole network space limits fig present phase spaces network topologies obtained simulations fourth condition network particular show phase space local dynamics patch graphics show case chain graph phase space patches different degrees characterized different dynamics fact patches show different behavior respect patches addition role patch degree see also position patches graph plays central role despite fact patches degree dynamics inside differs patches due different power dispersal rules two neighbors namely patches patches cause larger flux predators dispersal towards patches global effect presence three different dynamics one another one third one characterized oscillations regular amplitudes compare phase spaces standard phase space single patch model influence network topologies metapopulations dynamics chain grid patch patch ring star patch patch complete random patch patch figure power migration dynamics phase space network topology besozzi given fig right side also phase spaces fig graphics furthermore evidence oscillations characterized initial wider amplitude reduced time similarly dynamics patches grid graph phase space influenced number edges phase space identify two different types dynamics one patches three edges another one two connections star graph phase space dynamics endures patches apart number preys collapses attractor zero oscillations according dynamics established patch number predators fluctuates certain range dispersal patches basically condition patch represents center star becomes local area habitat dispersal occurs simulations ring complete graphs phase spaces show similar results cases patches graph degree two first configuration five second one leading regular oscillations almost constant amplitude results concerning last configuration random graph phase space show combination effects described particular dynamics patches differ depending degree patches moreover characterized highest degree high number incoming predators migrating four adjacent patches leads extinction preys similarly happens patch star graph also tested network topology three conditions listed cases results shown amplification power dispersal respect patch degree gives rise balance incoming migrating individuals leads comparable dynamics networks regular oscillations inside patch data shown network topologies colonization section compare six network topologies respect capability colonizing empty patches network contains starting patches contain complete model occupy peculiar position network recall work considering migration predators hence empty patches hereby assumed contain predators initial amount preys network set patches initialized complete model denoted test feature colonization consider four different initial conditions hereby denoted ick characterized characterized characterized characterized given network empty patches initialized chosen condition ick besides patches set initialized standard model communication constant equal one given chosen ick parameters given section type analysis expect determine features network topologies relevant respect colonization empty patches given initial condition conditions tested network fixed initial condition different sets considered following space limits present results simulations influence network topologies metapopulations dynamics briefly discuss results obtained analyzed conditions following graph preys represented solid lines predators represented dashed lines start considering network chain graph case present results obtained initial conditions considering three sets patches namely palv palv palv first case palv shown fig see power dispersal low time required predators reach patches highest distance allows initial uncontrolled growth preys subsequently undergo extinction soon predators enter patch delay local establishment population predators effect prevent formation dynamics effect shown hereafter common aspect network topologies concerning chain network evident condition initial amount preys inside empty patches higher case dynamics established four six patches hand initial conditions power dispersal sufficient colonize patches irrespectively numbers preys initially present empty patches position complete patch similar results chain network obtained second analyzed case palv shown fig third case palv data shown individuals individuals patch time patch time individuals individuals patch time patch time figure colonization chain topology palv initial conditions top left top right bottom left bottom right besozzi individuals individuals patch time patch time individuals individuals patch time patch time figure colonization chain topology palv initial conditions top left top right bottom left bottom right network topology grid graph show results obtained cases pblv fig left side pblv fig right side according position complete patches network topology see first case predators capable colonize patches directly connected patch directly connected however patches colonized second case higher degree complete patch allows colonization patches initial condition data shown tested cases pblv pblv patches directly connected respectively colonized predators network topology star graph show results obtained cases pclv fig left side pclv fig right side according position complete patches network topology see first case patches colonized high degree patch connected spreads predators patches thus preventing formation dynamics second case combined effect migration allows colonization patch directly connected performed simulations starting conditions cases higher value allows colonization every patch except patch independently initial position complete patch data shown contrary influence network topologies metapopulations dynamics individuals individuals patch time patch time figure colonization grid topology initial condition pblv left pblv right individuals individuals patch time patch time figure colonization star topology initial condition pclv left pclv right individuals individuals patch time patch time figure colonization ring topology pdlv initial condition left right besozzi assume pclv center star patches fully colonized independently considered initial condition network topology circular graph show results obtained cases pdlv fig left right sides respectively starting initial condition predators capable colonizing patches directly connected complete patch case also patch distance complete patch colonized results highlight particular another aspect marginal simulations stochastic nature communication process growth preys leads extinction preys patch patch drives local behavior oscillatory dynamics network topology complete graph show results obtained cases pelv fig left side pelv fig right side second case dynamics initially placed two patches predators colonize patches first case colonization empty patches fails effect stochastic noise combined low amounts predators turn caused fact higher number adjacent patches lower number predators persist inside patch simulations performed initial conditions patches always colonized higher values dispersal rules assure uniform spread predators throughout network thus flattens influence migration delay data shown network topology random graph show results obtained cases plv fig left side plv fig right side according position complete patches network topology see first case patches colonized predators similar results obtained placing complete model patch data shown second case patch colonized one path length connects initial complete patch holds patch distance equal similar reasons considering case initial condition complete model patch patch colonized predators data shown simulations performed initial condition patches colonized high amount preys initially occurring patches hand initial conditions power dispersal allows colonization patches data shown individuals individuals patch time patch time figure colonization complete topology initial condition pelv left pelv right influence network topologies metapopulations dynamics individuals individuals patch time patch time figure colonization random topology initial condition plv left plv right discussion fragmented habitats real metapopulations usually characterized complex network topologies paper analyzed six small topologies considered representative local areas structured habitat investigated influence degree position patch topology migration individuals well capability colonizing empty patches analysis suggests respect power migration section identify different behaviours depend two characteristics topology first level local behaviour inside patch influenced degree especially evident compare network topology described circular complete graphs topology described star graph first case nodes degree patches characterized similar regular oscillating dynamics second case critical node center star much higher degree nodes graph latter case patch likely undergo local modification initial dynamics due higher incoming migration individuals adjacent patches second level assuming case degree nodes equal also position patch topology matters instance seen network topology described chain graph nodes besides ones extremes chain degree local dynamics also influenced dynamics adjacent patches graph therefore hypothetical habitats exist many patches connected linear way results suggest length chain might negative role establishment maintenance local dynamics considering feature colonization section evidenced network topologies lack colonization due delay migrating predators respect uncontrolled local growth prey leads extinction preys prevention dynamics effectively measure strong power delay would interesting understand whether local growth preys controlled inducing death thus potentially allowing establishment oscillations besides aspect deserving investigations analysis evidenced colonization empty patches occurs easily patches adjacent patch initialized complete model highlights relevance position patch standard oscillations preys predators already besozzi settled beginning simulation indeed power colonization stronger circular complete networks position complete patch irrelevant spread migrating individuals throughout network results uniform weaker star network position complete patch primary importance spread migrating individuals throughout network strongly depends whether patch placed center tips star addition investigations presented work types analysis plan perform metapopulation systems concern instance study aspects considered paper migration colonization network topologies etc assuming local global dynamics population growth according logistic function moreover interesting issue might investigated synchronization local population dynamics considering establishment decay oscillations preys predators migration given network topology process colonization concerning use graphs relevant questions regard analysis dynamics respect graph properties different measures habitat connectivity centrality indexes context example star graph resemble notion hub node high degree typical network structure known robust random disturbances highly vulnerable deliberate attacks hubs another topic interest concerns fact various populations coexist common habitat distinct inter species dynamics different dispersal capabilities habitat cases like would interesting construct analyze different metapopulation models one target species according connections specific population dynamics comparing intersecting results obtained distinct network topologies common habitat derived way would possible determine locations habitat important species thus aid design natural reserve systems appropriate solution species terms maximal improvement dispersal reduction species isolation minimal spread disturbances diseases pathogens invasive species etc believe modeling approach opens interesting perspectives represent useful tool investigation wide range properties metapopulation systems expect applications model real cases characterized complex habitat networks patch possesses features quality occupancy connectivity different population dynamics aid achievement important results new perspective ecology references aittokallio schwikowski methods analysing networks cell biology briefings bioinformatics available http albert networks cell biology journal cell science available http albert statistical mechanics complex networks reviews modern physics available http berec techniques spatially explicit models construction simulation analysis ecological modelling available http influence network topologies metapopulations dynamics besozzi cazzaniga mauri pescini biosimware simulation environment biological systems accepted presentation jena germany besozzi cazzaniga pescini mauri seasonal variance system models metapopulations progress natural science available http besozzi cazzaniga pescini mauri modelling metapopulations stochastic membrane systems biosystems available http besozzi cazzaniga pescini mauri algorithmic bioprocesses chapter multivolume approach stochastic modelling membrane systems springer verlag available http bunn landscape connectivity conservation application graph theory journal environmental management available http cao gillespie petzold efficient step size selection simulation method journal chemical physics available http cazzaniga pescini besozzi mauri tau leaping stochastic simulation method systems hoogeboom rozenberg salomaa editors proc international workshop membrane computing lncs available http ciobanu editors applications membrane computing springerverlag berlin dorogovtsev mendes evolution networks advances available http physics dunning stewart danielson noon root lamberson stevens spatially explicit population models current forms future uses ecological applications available http fall fortin manseau brien spatial graphs principles applications habitat connectivity ecosystems available http gillespie exact stochastic simulation coupled chemical reactions journal physical chemistry available http hanski metapopulation dynamics nature available http hastings harrison metapopulation dynamics genetics annual review ecology systematics available http hastings wolin dynamics metapopulation ecology available http jansen dynamics two diffusively coupled populations theoretical population biology available http jansen lloyd local stability analysis spatially homogeneous solutions systems journal mathematical biology available http besozzi levins demographic genetic consequences environmental heterogeneity biological control bulletin entomological society america minor urban framework evaluating landscape connectivity conservation planning conservation biology available http moilanen spomsim software stochastic patch occupancy models metapopulation dynamics ecological modelling available http murray mathematical biology introduction new york newman structure function complex networks siam review available http computing membranes journal computer system sciences available http membrane computing introduction berlin rozenberg salomaa editors oxford handbook membrane computing oxford university press pescini besozzi mauri zandron dynamical probabilistic systems international journal foundations computer science available http pescini besozzi zandron mauri analysis simulation dynamics probabilistic systems pierce carbone editor proc international workshop dna computing lncs london canada available http strogatz exploring complex networks nature available http taylor metapopulations dispersal dynamics overview ecology available http travis dytham evolution dispersal metapopulation spatially explicit model proceedings royal society london series biological sciences available http urban keitt landscape connectivity perspective ecology available http urban minor treml schick graph models habitat mosaics ecology letters available http weisser jansen hassell effects pool dispersers systems journal theoretical biology available http
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dec ieee ieee gsm lte gps cdma inria max min min max ord ord ord magnitude response normalized frequency magnitude response normalized frequency magnitude response normalized frequency magnitude response normalized frequency magnitude response normalized frequency magnitude response normalized frequency magnitude response normalized frequency magnitude response normalized frequency magnitude response normalized frequency magnitude response normalized frequency magnitude response normalized frequency magnitude response normalized frequency phase response rad normalized frequency group delay samples normalized frequency amplitude sample number bogatyrev chebyshev representation rational functions mohan singh biswas generalized synthesis design symmetrical multiple passband filters progress electromagnetics research vol lee sarabandi design microwave filters using frequency transformation ieee trans microwave theory vol lunot bila seyfert optimal synthesis microwave filters ieee int microw symp jun deslandes boone iterative design procedure synthesis generalized filters microw vol macchiarella synthesis multiband prototype filters using algorithm ieee microwave wireless components letters vol may approximation problem electrical filters birkhauser advances microstrip filters cambridge university press bogatyrev computations moduli spaces computational methods function theory bogatyrev extremal polynomials riemann surfaces springer monographs mathematics isbn print veidinger numerical determination best approximations chebyshev sense numer vol boyd vandenberghe convex optimization cambridge university press fuchs chebyshev approximation sets several components brannan clunie aspects contemporary complex analysis academic press cauer theorie der linearen wechselstromschaltungen becker und erler leipzig akademie berlin vii
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bloofi multidimensional bloom filters adina crainiceanua daniel lemireb licef naval academy usa research center teluq university quebec canada sep abstract bloom filters probabilistic data structures commonly used approximate membership problems many areas computer science networking distributed systems databases increase data size distribution data problems arise large number bloom filters available need searched potential matches example federated cloud environment cloud provider could encode information using bloom filters share bloom filters central coordinator problem interest whether given element sets represented bloom filters existing sets contain given element problem solved constructing bloom filter union sets instead effectively multidimensional bloom filter problem given element wish receive list candidate sets element might solve problem consider alternatives firstly naively check many bloom filters secondly propose organize bloom filters hierarchical index structure akin tree call bloofi finally propose another data structure packs bloom filters way exploit parallelism call theoretical experimental results show bloofi provide scalable efficient solutions alternatives search large number bloom filters keywords bloom filter index multidimensional bloom filter federated cloud data provenance introduction bloom filters filters need searched find sets containing particular objects bloom filters used efficiently check whether object likely set match whether object definitely set match false positives possible false negatives due efficiency compact representation flexibility allowing space false positive probability bloom filters popular representing diverse sets data used databases distributed systems web caching network applications example google bigtable apache cassandra use bloom filters reduce disk lookups data digital data increases size distribution applications generate large number work motivated highly distributed data provenance applications data tracked created modified multiple sites participating application site maintaining data cloud environment bloom filters maintained individual site shared central location piece data need find sites holding data thus may need search large number bloom filters stored central location corresponding author tel fax email addresses adina adina crainiceanu lemire daniel lemire preprint submitted information systems indexing bloom filters different indexing generic objects improve search time one level indirection elements searched objects directly indexed index structure particular bloom filter compact representation underlying set elements question interest query given particular element bloom filter underlying sets contain element query subject element september objects indexing searching bloom filters creating traditional index structures hash indexes trees etc distributed versions directly apply case indexing bloom filters keys given site bloom filter significant work using bloom filters various applications developing variations bloom filters counting filters support deletions bloom filter compressed bloom filters used web caching stable bloom filters eliminate duplicates streams spectral bloom filters extend applicability bloom filters bloom filter mbf use perelement probabilities yet attempts accelerate queries many bloom filters call multidimensional bloom filter problem even though problem closely related signature file methods see section one seeks index setvalue attributes solve problem propose bloofi bloom filter index hierarchical index structure bloom filters bloofi provides probabilistic answers allmembership queries scales tens thousands bloom filters probability false positives low bloofi order tunable parameter provide logd search cost number bloom filters indexed bloofi also provides support inserts deletes updates logd cost requires storage cost designing bloofi take advantage fact bitwise bloom filters length constructed using hash functions also bloom filter resulting bloom filter represents union sets represented individual bloom filters property allows construct tree leaf levels indexed bloom filters root level bloom filter represents elements system tree used prune search space eliminate bloom filters candidates matches processing queries performance evaluation shows bloofi performs best false positive probability union bloom filter bloom filter union indexed bloom filters low provides logd search performance cases storage cost logd maintenance cost bloofi could used whenever large number bloom filters use hash functions need checked matches bloom filters constructed bitmaps vector booleans bitmaps exploit parallelism processor compute bitwise bits using single instruction use parallelism bloofi optimize construction data structure however also designed alternative data structure designed specifically exploit parallelism henceforth though scalable bloofi fast number bloom filters moderate article extended version bloofi hierarchical bloom filter index applications distributed data provenance published international workshop cloud intelligence paper completely revised new version introduces additional data structure flatbloofi new implementation bloofi improves performance order magnitude new performance evaluation rest paper structured follows section describes distributed data provenance application bloofi section briefly reviews concept bloom filter section introduces bloofi hierarchical index structure bloom filters section introduces maintenance algorithms theoretical performance analysis section introduces data structure multidimensional bloom filter problem designed exploit parallelism section shows experimental results discuss related work section conclude section motivation provenance application distributed data section describe distributed data provenance application motivated work bloofi let assume multinational corporation hundreds offices geographically distributed locations sites around world interested tracking documents produced used within corporation document given universally unique identifier uuid stored local repository cloud environment documents sent another location site received locations multiple documents bundled together create new documents therefore identified new uuids documents decomposed smaller parts become documents events important provenance document recorded repository site generating event events stored rdf triples scalable cloud triple store rya data modeled directed acyclic graph dag labeled edges event names nodes document uuids documents travel sites dag fact distributed machines cloud environment site also hundreds geographically distributed locations data provenance problem interested solving finding events document uuids form provenance path given uuid ancestors given node distributed graph storing data even uuids location centralized place feasible due volume speed documents generated globally fully distributed data structures chord require even communication messages centralized solution increasing latency bandwidth consumption also feasible due volume speed documents generated globally moreover local regulations might impose restrictions data stored however since global locations belong corporation data exchange data tracking must made possible without information location uuid provenance query uuid must sent sites site determine local part provenance path return however provenance might contain new uuids new query needs sent site new uuid connected original uuid new uuids found recursive process could consume significant bandwidth latency geographically distributed system minimize number unnecessary messages sent determine full provenance object local site maintains bloom filter uuids local system updates bloom filter periodically propagated centralized location headquarters example since bloom filters compact representations underlying data less bandwidth consumed bloom filters sent versus sending actual data central location bloofi index constructed bloom filters every time provenance query uuid made bloofi index used quickly determine sites might store provenance information given uuid query load high bloofi index single location affects performance availability multiple bloofi indexes could constructed several locations structure stored disk expect many requests values present helpful auxiliary data structure quickly dismiss requests bloom filters serve purpose bloom filter bit array bitmap length constructed using set hash functions empty bloom filter bits add element filter hash functions maps new element position bit array bit position turned check whether element member set represented bloom filter hash functions applied test element resulting positions test element set probability positions bloom filter matches test element test element might set might true positive false positive size bloom filter probability false positives pfalse returned pfalse assuming elements filter probability minimal probability pfalse lowered increasing size bloom filter fixed number elements probability goes zero exponentially rest paper assume bloom filters indexed length use set hash functions indexing bloom filters given collection bloom filters given query value want find bloom filters match interested case relatively bloom filters sequential search potentially inefficient bloofi hierarchical bloom filter index bloofi bloom filter index based following idea construct tree leaves tree bloom filters indexed parent nodes bloom filters obtained applying bitwise child nodes process continues root reached index property bloom filter tree represents union sets represented bloom filters rooted node consequence object matches bloom filter matches bloom filters path leaf root conversely particular bloom filter bloofi match object match entire rooted node using bloofi membership query starts first querying root bloom filter match bloom filters checking presence value set similar data structure expensive especially data queried object none indexed sets contain object negative answer returned root match object query proceeds checking child bloom filters matches object query continues path bloom filters matching object leaf level reached balanced tree height bloofi logarithmic number bloom filters indexed step query process goes one level tree best case query negative answer answered constant time check root query positive answer answered logarithmic time however multiple paths index followed query process query time increases section introduces heuristics bloofi construction number misleading paths bloofi reduced similar bloom filters many possible implementations bloofi tree similar binary search trees avl trees trees etc due flexibility allowed balanced tree number child pointers higher two implement bloofi like trees start order parameter node maintains child pointers nodes root node bloofi stores one value different general search trees leaves value bloom filter indexed nodes value obtained applying bitwise values child nodes throughout paper use usual definitions tree node tree root leaf depth node number edges root node height tree maximum depth node tree sibling parent fig shows example bloofi index order internal node child pointers leaf level tree contains original bloom filters indexed bloofi index identifiers node identifiers shown ease presentation also used practice updates deletes identify node needs updated deleted see section section details rest paper often use node refer node identifier next higher level values obtained applying bitwise values children nodes value node bitwise values nodes process continues root reached use following notation given node parent written ordered set children written number children written convention figure bloofi tree order zero leaf node children node node corresponding bit array indicated let bitwise operation node leaf bit array aggregation bit array children individual bit values bit array accessed since bloofi balanced tree internal node least child nodes order tree height bloofi index order blogd number bloom filters index search search algorithm algorithm returns identifiers bloom filters match given object subtree rooted given node first checks whether current node value matches object line none bloom filters match object empty set returned line current node match object either leaf case returns identifier line inner node case findmatches function called recursively child nodes lines example consider query object value bloofi tree figure findmatches function algorithm invoked arguments root algorithm findmatches node bloofi maintenance value identifiers leaves introduce algorithms inserting deleting updating bloom filters tree rooted node bloom filters matching object node matches object return empty set else check descendants match return else node leaf return identifier return else leaf check descendants returnlist findmatches return returnlist insert ideally inserting bloom filters would like keep partitions overlap different partitions small would like similar bloom filters grouped together much possible conversely would like bloom filters different partitions different possible problems commonly minimum graph bisection problem though sometimes approximated efficiently leave formal investigation problem future work use heuristic algorithm inserting algorithm finds leaf close input bloom filter given metric space inserts new bloom filter next leaf intuition similar bloom filters improve search performance distance metric use hamming distance count number bits differ computed quickly computing cardinality bitwise exclusive two bit arrays along fast functions count number resulting words java could consider distance metric experiment cosine jaccard metrics section new bloom filter inserted first updating value current node computing bitwise value filter inserted since node recursively calling insert function child node similar new value line similar leaf node located new leaf created new bloom filter line inserted sibling node calling insertintoparent function algorithm function takes parameters new node newentry similar node node insert newentry sibling node number children parent still insert complete overflow occurs node splits lines newly created node returned function splits could occasionally propagate root level case new root created height bloofi tree increases line example consider bloofi tree fig assume insert bloom filter value new node inserted child node hamming distance new node nodes line algorithm algorithm checks whether value root matches simplicity presentation assume one hash function used bloom filters function mod elements underlying set integers since root matches queried object search proceeds invoking findmatches function child nodes first child node node match queried object findmatches returns second child node root node matches search continues next lower level node matches queried object leaf findmatches function returns identifier leaf match queried object findmatches call returns recursive call node returns value finally call root returns result query search cost complexity search process given number findmatches invocations best case bloom filters matching given object number bloom filters checked matches root find leaflevel bloom filter matches query number findmatches invocations logd best case one path followed node children checked find one matches worst case since maximum number nodes bloofi tree search cost nodes need checked matches algorithm node algorithm insert newbloomfilter node distance function bit arrays rooted given node null pointer new child split occurred node leaf direct search new filter place value node contain new filter newbloomfilter similar child insert find child minimizing newbloomfilter newsibling insert newbloomfilter split return null newsibling null return null else split check whether new root needed null split create new root newroot new bfinode create new node null node newsibling root newroot newroot newroot return null else newsibling insertintoparent newsibling node insertintoparent newentry node parent node pointer null pointer new child split occurred newentry node overflow return null else split new bfinode move last children update parent information children children values compute children values return return newsibling node root split else node leaf need insert parent node newbloomfilter newleaf new bfinode newbloomfilter new leaf parent node newsibling insertintoparent newleaf node return newsibling node leaf figure bloofi tree insert split let assume node chosen closest node needs split resulting bloofi tree shown fig theorem gives cost insert algorithm bloofi cost metric use measure performance operation bloofi number bloofi nodes accessed operation either bloom filter value operation parent children pointers node sibling redistribution possible entries redistributed parent information moving nodes updated bloom filter values parent node sibling node way root updated lines redistribution possible parent node merges sibling giving entries sibling lines bloom filter value sibling node updated delete procedure called recursively parent node occasionally delete propagates root one child remains root root deleted height tree decreases lines example assume node value deleted fig resulting tree deletion node redistribution shown fig theorem insert cost number bloofi nodes accessed insert operation bloofi logd order bloofi index number bloom filters indexed roof following components part insert operation theorem delete cost number bloofi nodes accessed delete operation bloofi logd order bloofi index number bloom filters indexed values nodes path root new leaf updated reflect newly inserted bloom filter updating value node means computing newly inserted bloom filter old new bloom filter values need accessed cost update therefore constant height bloofi tree blogd level perform constant amount work total cost update logd roof reference node deleted parent bloofi tree constant cost operation values nodes deleted node root need recomputed total cost logd occasionally redistribute merge needed cost worst case merge delete propagates root worst case cost merge logd follows cost delete operation logd level tree search similar child node performed line cost search total cost due search placement new leaf logd subjected many insertions deletions though bloofi tree remains balanced could partition bloom filters per hamming distance could degrade quality performance bloofi could diminish instances could become necessary reconstruct bloofi data structure cost split since children node worst case split propagates root height tree logd worst case cost split operations logd update object insertions underlying set lead updates bloom filters expect update operation bloofi quite frequent instead treating bloom filter update delete followed insert use update bloofi tree becomes inefficient routing due updates many false positives search bloofi tree reconstructed scratch batch mode algorithm shows update algorithm algorithm takes parameters leaf node corresponding updated value new bloom filter value node bloom filters path leaf root updated new value three points see cost insert operation logd delete delete algorithm algorithm deletes given node bloofi index procedure first invoked leaf deleted argument pointer leaf parent deleted parent node underflowing least children left bloom filter values nodes path parent node root bitwise remaining children line delete procedure terminates underflow lines parent node tries redistribute entries algorithm delete childnode parent node parentnode reference node parent childnode whether tree height needs reduced parentnode root root null return null check underflow parent try redistribute first sibling sibling parentnode sibling remove children sibling even number children insert new children parentnode update information nodes moved figure bloofi tree delete redistribute value nodes involved root children value recomputevaluetotheroot parentnode else sibling move children parentnode sibling update information nodes moved sibling value childrenvalue parentnode delete parentnode redistribute else underflow value bloom filters root recomputevaluetotheroot parentnode algorithm update leaf newbloomfilter values path leaf root node leaf repeat newbloomfilter node node null figure bloofi tree update example bloofi tree fig assume update value node values nodes path node root resulting tree shown fig checking membership bloom filter equivalent checking value bits random locations bitmap though fast operation exploit parallelism processor ability several bitwise operations one instruction let assume processor propose new approach stores data corresponding bloom filters packed data structure bloom filter backed bitmap place construct single array integers length henceforth array first integer corresponds first bit bitmaps thus value ith bit jth bitmap value jth bit ith integer array given bloom filters create arrays multiple bits unused one arrays organize using following data structures theorem update cost number bloofi nodes accessed update operation bloofi logd order bloofi index number bloom filters indexed improving pruning efficiency bloofi node value bitwise children values total number objects underlying sets indexed bloofi increases probability false positive results returned bloom filters higher levels tree increases worst case bits bloofi nodes higher levels tree could one leads decreased pruning efficiency higher levels tree false positive paths followed search improve upon number bloom filters need checked matches query propose following heuristic insert procedure split node bits set one even node full could stop splitting little early avoids creating multiple levels tree bits set one experimental results section show search cost indeed improved using heuristic root level value bits set one effectively bloofi viewed forest instead tree alternatively could dynamically change size bloom filters false positive probability root reaches case application allows could reconstruct base bloom filters lower false positive probability reconstruct bloofi tree index construction time bloom filters seconds experiments periodic reconstruction index viable solution maintain arrays data index bloom filters use array bits exactly set true indicates index locations use thus bloom filter deleted index new one inserted reuse space use hash table maps bloom filter identifiers internal index values range also maintain array identifiers length gives identifier bloom filter stored given index combined together hash table array identifiers provide index bloom filter identifiers index locations queries given hash functions map given value index locations range flatbloofi array retrieve corresponding integers compute aggregate iterate bits value true one corresponds matching bloom filter use array identifiers recover corresponding bloom filter identifiers thus bloom filters access integers arrays iterating set bits done quickly using fast functions java parallelism bloofi keeps bloom filters adds new aggregated bloom filters accelerate queries however worst case scenario bloofi may need check many bloom filters case bloofi may faster could even slower naive approach merely checks every bloom filter insertion inserting new bloom filter using bit array first seek available index none found create new array integers thus effectively making available new index positions bit array array identifiers hash table mapping index values updated finally iterate set bits bitmap new bloom filter set corresponding bits array using operation size bloom filters different underlying data distributions bloofi different similarity metrics used bloofi construction different order values also compare bloofi performance naive case bloom filters searched linearly without index show cases bloofi achieves logarithmic search performance low maintenance cost regardless underlying data distribution deletion deleting bloom filter use hash table recover index using identifier key removed hash table corresponding bit set false two possibilities experiments setup implemented bloofi java ran experiments using oracle jdk version bloom filters use implementation provided skjegstad modified use faster hashing efficient bitset implementation experiments run workstation intel xeon processor ghz cores test machine ram use parallelism data structures ram bloom filter stored alone array array removed also remove corresponding bits well corresponding entries array identifiers scan values hash table deduct index entries exceeding index deleted bloom filter otherwise array unset bits corresponding bloom filter using operation keep copy original bloom filter need update every single component array performance metrics performance metrics use search number bloom filters checked find one matching queried object averaged searches believe compaction approach provide reasonable performance memory usage context deletions insertions frequent worst case scenario however many deletions could arrays indexing bloom filters guard inefficiencies would need aggressive compaction strategies leave future work note could accelerate queries replacing many arrays single array containing words bits would improve memory locality however would make compaction expensive search time average time milliseconds find bloom filters matching queried object storage cost space required store bloom filters associated index structure applicable bloofi estimate cost number bytes bloom filter multiplied number nodes bloofi tree including leaves storage cost estimated number bytes bloom filters multiplied number bloom filters rounded multiple uses longs storage naive case storage cost estimated number bytes bloom filters multiplied number bloom filters update updating bloom filter easiest corresponding array set corresponding bits insertion experimental evaluation maintenance average number bloofi nodes accessed insert delete update operation evaluate bloofi search performance maintenance cost different number maintenance time average time milliseconds insert delete update operation parameters varied described section bloom filter bit array length constructed using set hash functions practice constructing bloom filter one specify length number hash functions unlikely average engineer would know values pick rather one specifies expected maximal number elements stored nexp possibly large number well desired probability false positive one reasonably expect engineer able set values domain knowledge number hash functions size bitmaps computed using following formulas nexp experiments vary following parameters parameter value number bloom filters indexed bloofi order bloom filter size bits elements bloom filter desired probability false positives construction method similarity measure data distribution iterative hamming nonrandom table default values parameters run experiment times report averages last runs since values integers picked hash functions random form mod odd integer defining hash function found choice gave good performance case number bloom filters indexed bloofi order performance results varying number bloom filters indexed fig shows increase search time increases bloofi naive case note logarithmic scale axes bloofi increase search time logarithmic long false positive probability pfalse root less one increase higher logarithmic due high value pfalse high levels tree however even filters bloofi still performs orders magnitude better naive case difference naive case bloofi big bloom filters checked likely search time depends number bloom filters checked also locality nodes memory see curves bloofi intersect small numbers bloom filters performs better due superior memory locality exploitation parallelism however number bloom filters increases bloofi performs better due superior pruning abilities bloofi using bulk construction leads improved search performance since global sort bloom filters performed bulk insert incremental construction greedy might lead optimal placement however cost sorting implementation leads high index construction time bulk construction evaluate effect update bloofi introduced section performed experiments bloofi tree built incrementally using size bits bloom filters indexed varied indirectly specifying nexp number elements bloom filter indexed desired probability false positives bloom filters indexed index construction method iterative insert bloom filters one one using algorithm section bulk bulk construction first sort bloom filters first bloom filter one closest empty bloom filter second filter closest first bloom filter construct bloofi tree always inserting next leaf similarity measure measure used define closeness insert consider hamming cosine jaccard distances data distribution nonrandom nonoverlapping ranges bloom filter bloom filter contains integers range random overlapping ranges data bloom filters bloom filter contains random integers randomly assigned range experiment vary one parameter use default values shown table rest flat bloofi naive number bloom filters indexed search time ideal bloofi naive storage cost search search time flat bloofi naive number bloom filters indexed search number bloom filters indexed storage cost figure varying number bloom filters half elements bloom filter rest elements inserted bloom filter bloofi updated perform similar experiment naive case updates curves fig show search time final data structure expected naive updates normal curves almost identical since properties data structures affected updates however bloofiau bloofi curves also almost identical shows update maintains search performance bloofi tree shows using heuristic increases search performance bloofi index false positive probability high levels tree high fig shows increase storage cost bloofi naive number bloom filters indexed increases cases storage cost increases linearly storage cost lowest naive case extra information besides actual bloom filters maintained small overhead sometimes rounds space multiples take advantage bitlevel parallelism storage cost bloofi also quite low number nodes bloofi tree less storage cost bulk construction slightly higher incremental construction bloofi constructing tree always inserting leaf leads general skinnier trees levels nodes fig shows cost terms time maintaining bloofi naive data structure maintenance cost naive negligible bloom filters maintained list maintenance cost increases slightly number bloom filters cost different operations depends mainly size bloom filters number bits turned depend much number bloom filters bloofi trend increasing insert delete nodes bloofi tree get impacted insert delete cost updates increase root becomes one split use updates height tree increase find relative cost insert delete update bloofi quite interesting bloofi update cheapest operation time number bloom filters accessed see fig since values nodes leaf root need updated new value however fig shows similar trends search average number bloom filter nodes accessed search search performance metric used naive data strictures bloom filters checked search ideal case bloofi exactly one path root leaf followed search search approximately logl node children search increases logarithmically experiments increase search logarithmic long false positive probability pfalse root less one increase higher logarithmic due high value pfalse high levels tree however even filters bloofi still performs two orders magnitude better naive case size bloom filters increases search cost decreases ideal cost shown fig evaluate effects heuristic introduces section run experiment without using heuristic always splitting root even bits set bloofi nodes bits search search time heuristic used lower heuristic used search number bloom filters indexed maintenance time average nodes accessed average nodes accessed operation time number bloom filters indexed maintenance bloofi order maintenance figure maintenance cost bloofi insert operation expensive delete becomes expensive time increased number indexed bloom filters reason difference delete operation fewer nodes accessed need search place tree work node need values node path leaf root children value insert values nodes get updated newly inserted bloom filter inserting sparse bloom filter words need change fast deleting know bit changed set zero takes longer bloom filters would dense possible deletions could faster insertions updates know new bloom filter differs old one compute difference xor since longer old one set bits bloom filter set running time close insert see fig comparing bloofi bloofi updates cheaper bloofi needs new value bloom filters inserts faster need search compute distances tree best place insert new node deletes bloofi faster number bloom filters pfalse root cost depend much number bloom filters becomes faster large number bloom filters indexed tributions needed update increase root bloofi tree becomes one root split case height bloofi tree increase increased number indexed bloom filters varying bloofi order fig fig show search time search order bloofi increases search bfcost search time increase increases since search cost proportional logd constant function log convex minimum value best search performance achieved bloofi tree low order search obtained experiments close ideal search cost full trees children per node shows algorithms tree construction perform well many nodes outside path root answer leaf checked search bulk construction performs better incremental construction due global sort bloofi lines almost identical shows update maintains performance bloofi tree storage cost decreases order fig number nodes bloofi tree order higher order lower overhead storing constructed bloofi tree storage cost lower higher orders search cost higher fig search storage cost expect search performance important storage practice storage cost bloofi similar naive case even worst case twice much space needed bloofi needed naive case believe bloofi trees low order used practice storage cost bulk construction slightly higher incremental construction always inserting leaf leads maintenance bloofi average number bloom filters accessed maintenance operation shown fig maintenance increases logarithmically fig expected theorems insert higher delete place new node needs found insert update cost lowest use updates splits merges search time search bloofi bloofi bloofi order search time storage cost bloofi naive bloofi order search bloofi order storage cost figure varying bloofi order pfalse increases pfalse root values tested experiment bloofi performance close ideal case skinnier taller trees nodes fig shows variation maintenance order bloofi tree insert delete costs logd increase update cost logd decreases surprising result shown fig search time naive case bloofi decreasing experiment even search bfcost increasing main reason memory locality increases size bloom filters decreases leading better memory locality properties efficient exploitation bitlevel parallelism benefits low memory footprint consistent results shown fig second reason decrease search time cases number hash functions used bloom filters decreases increases decreases experiment fewer hash functions used fewer bits bloom filters need checked matches contributes faster search time varying bloom filter size vary bloom filter size varying number expected elements bloom filter nexp nexp fig shows search variation bloom filter size system total elements filters elements search bloofi always naive cost decreases ideal cost size bloom filters increases false positive probability high level nodes bloofi decreases fact logd search cost achieved soon pfalse root even pfalse close experiments size search time fig also decreases bloofi pfalse higher levels reduced search time naive case almost constant small bloom filter sizes search time naive case slightly lower using bloofi likely due memory locality search time lower bloofi pruning capabilities bloofi reduced due efficient bit level parallelism exploited flatbloofi size bloom filters increases memory locality reduced trend search time upward trend downward bloofi varying number elements fig shows search time varies increase number elements indexed bloom filters size bloom filters remains constant provides best performance case benefits memory locality parallelism performance almost constant regardless actual number elements bloom filter search time bloofi increases slightly long pfalse root even number elements system ten times larger expected number elements bloom filter elements system nbelexp bloofi performance degrades pruning capabilities reduced pfalse root search bloofi varies similar way shown varying probability false positives fig shows search bloofi increases increase desired probability false positives bloom filters indexed expected slowly increases implicitly pfalse increases pruning capabilities bloofi reduced ideal bloofi naive search search time search flat bloofi naive hamming cosine jaccard bloom filter size bits bloom filter size bits search filter size number bloom filters indexed search time filter size search metric figure flat bloofi naive desired false positive probability search search time ideal bloofi naive search time search desired false positive probability search time flat bloofi naive number elements bloom filters search time figure filters set experiments nonrandom case overlap sets represented different bloom filters random case could overlap sets expect real scenario random case common search cost shown fig search time shown fig expected performance random case little worse nonrandom case since multiple bloom filters might match queried object however overlap ranges nonrandom case enough lead substantial increase search cost search time nonrandom case even number results indeed larger nonrandom case search performance similar distributions used shows bloofi robust underlying data distribution varying similarity metric bloofi construction use similarity metric measure similarity two bloom filters metric determines location bloom filters bloofi section experiment several metrics hamming xor jaccard cosine number bit array search similar metrics fig jaccard used search cost little lower differences small might due chance similar trend obtained search time shown fig search time lowest jaccard highest hamming experimental results conclusion varying underlying data distribution use following distributions underlying data bloom filters nonrandom insert bloom filter integers range random insert bloom filter integers randomly chosen random range assigned bloom filter actual number elements bloom experimental evaluation shows bloofi search cost increases logarithmically gets worse false positive probability root optimal performance size bloom filters based estimate total number elements entire system false positive number bloom filters indexed search time metric naive search time hamming cosine jaccard search search time number bloom filters indexed number bloom filters indexed search data distribution search time data distribution figure probability root even close search cost logd search time increases low number bloom filters search time lower bloofi search costs bloofi increase order since search cost logd low order preferred bloofi storage cost bloofi bloofi insert cost delete cost logd update cost logd bulk construction gives slightly better search results since trees constructed bulk construction skinnier however differences performance bulk incremental construction small cost sorting first step bulk construction implementation operation expensive experimental results show bloofi algorithm insertion produces tree close tree obtained using global ordering bloom filters distance metric used compare closeness bloom filters big effect search performance resulting bloofi tree jaccard distance seems lead best search performance based bloom filters uses data structure similar mullin uses bloom filters reduce cost semijoins distributed databases applications bloom filter use bloom filters directly search large number bloom filters trees inspired implementation bloofi tree however node bloofi one value children node represent completely disjoined sets multiple paths might followed search closely related idea multidimensional bloom filter problem dates back superimposed codes descriptor files descriptors effectively bit arrays searched large files hierarchical data structure constructed descriptions applied attributes descriptors called signatures possible value mapped fixed number bit positions set true value present sets constructed setting bit positions corresponding values implementation idea resembles bloofi signatures organized similar tree structure using hamming distance aggregate similar signatures primarily used index attributes problem differs provided bloom filters distributed setting must quickly locate bloom filter matches given query related work part existing work related bloom filters concerned extending improving bloom filters deal problem searching large set bloom filters applications bloom filters web caching use multiple bloom filters number general small linear search bloom filters performed introduces xml filtering system approach similar wordparallel wpbs approach used scanning signatures except data structure inmemory supports moderately fast deletion require dedicated hardware also similar bitsliced signature files apply text indexing conclusions future work cohen matias spectral bloom filters proceedings acm sigmod international conference management data sigmod pages new york usa acm comer ubiquitous acm comput june crainiceanu bloofi hierarchical bloom filter index applications distributed data provenance proceedings international workshop cloud intelligence pages new york usa acm crainiceanu linga machanavajjhala gehrke shanmugasundaram efficient robust range index structure proceedings acm sigmod international conference management data sigmod pages new york usa acm deng rafiei approximately detecting duplicates streaming data using stable bloom filters proceedings acm sigmod international conference management data sigmod pages new york usa acm deppisch dynamic balanced signature index office retrieval proceedings annual international acm sigir conference research development information retrieval sigir pages new york usa acm dutta bhattacherjee narang towards intelligent compression streams biased reservoir sampling based bloom filter approach proceedings international conference extending database technology edbt pages new york usa acm fan cao almeida broder summary cache scalable web cache sharing protocol trans ficara giordano procissi vitucci blooming trees structures data representation proceedings ieee international conference communications pages may gong qian yan zhou bloom xml packets filtering millions path queries proc icde pages lee kim patel efficient signature file methods text retrieval ieee trans knowl data jun lee lochovsky hybrid machine large databases ieee trans jan lemire javaewah compressed alternative java bitset class version https last checked cui xia wang scalable data center multicast using bloom filter proceedings ieee international conference network protocols icnp pages washington usa ieee computer society mitzenmacher compressed bloom filters proceedings twentieth annual acm symposium principles distributed computing podc pages new york usa acm mitzenmacher upfal probability computing randomized algorithms probabilistic analysis cambridge university press new york usa mullin optimal semijoins distributed database systems ieee trans softw may pfaltz berman cagley partialmatch retrieval using indexed descriptor files commun acm introduced bloofi hierarchical index structure bloom filters taking advantage intrinsic properties bloom filters bloofi reduces search cost membership queries thousands bloom filters efficiently supports updates existing bloom filters well insertion deletion filters experimental results show bloofi scales tens thousands bloom filters low storage maintenance cost extreme worst case bloofi performance similar using index search cost vast majority scenarios bloofi delivers close logarithmic performance even false positive probability root close less one fewer bloom filters found alternative designed exploit parallelism fared better could pursue advanced applications example bloofi could applied current data moving window needs maintained window interest multiple days separate bloom filters constructed data collected day old bloom filters deleted bloofi new filters inserted objects current window represented bloofi index acknowledgements special thanks crainiceanu insightful discussions developing bloofi rapp skene cooper providing motivation applications work references aguilera golab shah practical scalable distributed proc vldb ahuja roberts processor partial match retrieval using superimposed codes proceedings annual symposium computer architecture isca pages new york usa acm bloom hash coding allowable errors commun acm july broder mitzenmacher network applications bloom filters survey internet pages chang dean ghemawat hsieh wallach bigtable distributed storage system structured data osdi pages chang dean ghemawat hsieh wallach burrows chandra fikes gruber bigtable distributed storage system structured data acm trans comput june punnoose crainiceanu rapp rya scalable rdf triple store clouds proceedings international workshop cloud intelligence pages new york usa acm roberts retrieval via method superimposed codes proceedings ieee dec kent ramamohanarao multikey access methods based superimposed coding techniques acm trans database skjegstad bloom filter implementation written java version https last checked stoica morris karger kaashoek balakrishnan chord scalable lookup service internet applications proceedings conference applications technologies architectures protocols computer communications sigcomm pages new york usa acm chen ooi towards elastic transactional cloud storage range query support proc vldb zobel moffat ramamohanarao inverted files versus signature files text indexing acm trans database
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neuromorphic hardware architecture using neural engineering framework pattern recognition runchun wang chetan singh thakur tara julia hamilton jonathan tapson van schaik marcs institute university western sydney sydney nsw australia present hardware architecture uses neural engineering framework nef implement largescale neural networks field programmable gate arrays fpgas performing pattern recognition real time nef framework capable synthesising cognitive systems subnetworks first present architecture proposed neural network implemented using numbers demonstrate routine computes decoding weights using online pseudoinverse update method opium parallel distributed manner proposed system efficiently implemented compact digital neural core neural core consists neurons instantiated single physical neuron using approach proof concept combined identical neural cores together build handwritten digit recognition system using mnist database achieved recognition rate system implemented fpga process million digits per second architecture limited handwriting recognition generally applicable extremely fast pattern recognition processor various kinds patterns speech images millions pattern recognitions real time nef first introduced framework capable building large systems subnetworks standard neural structure first layer contains input neurons second layer hidden layer consists large number neurons third layer output layer consists linear neurons nef used construct spaun first brain model implemented software capable performing cognitive tasks eliasmith demonstrates nef powerful tool synthesising cognitive systems previously presented compact neural core architecture specifically fpga implementation large nef networks wang paper present application uses neural core build pattern recognition systems outline paper follows section introduces basic concepts nef algorithm theory presented section hardware implementation presented section performance different design choices thoroughly compared section section compare work solutions discuss future works keywords neural engineering framework pattern recognition pseudo inverse mnist neuromorphic engineering materials methods introduction background neural networks proved powerful tools real world tasks pattern recognition classification regression prediction however high computational demands ideally suited modern computer architectures constraint far often prohibited use applications need control interactive robotic systems hand scientists developing hardware platforms optimised neural networks past two decades vogelstein boahen pfeil wang however systems capable synthesising neural networks real world tasks subnetworks therefore suitable pointed tapson tapson section review theoretical framework present generic hardware architecture uses neural engineering framework nef eliasmith anderson implement neural networks fpgas capable processing figure typical nef network stimulus encoded large number nonlinear hidden layer neurons using randomly initialised connection weights output system linear sum weighted spike trains hidden neurons trained using training dataset subsequently validated using test dataset proposed digit recognition system feed forward neural network consisting input layer neurons pixels hidden layer neurons ten output layer neurons input layer neurons connected hidden layer neurons using randomly weighted connections hidden layer neurons also connected neurons using connections weights calculated using pseudoinverse operation digit recognition system single input digit pixels mapped onto layer input neurons refer vector img dimension img matrix multiplied matrix dimension resultant vector referred vin dimension thus given figure tuning curves maps input stimuli spike rates clarity figure shows tuning curve neurons neuron neural layer distinct tuning curve value vin sum randomly weighted pixels stimulus corresponding neuron hidden layer neuron hidden layer responds vin value according distinct tuning curve figure output hidden layer neurons input digit collected matrix referred dimension finally response output layer neuron given typical nef system encodes input stimulus spiking rate neurons heterogeneous population decodes desired function linearly combining responses neurons topology nef network illustrated figure nef network performs three tasks calculate desired function encoding encoder fixed random weight hidden layer neuron multiplies input stimulus weight firing rate individual neurons nonlinear function input stimulus weighted random weights parameters neurons also randomised neuron hidden layer exhibits distinct tuning curve example tuning curves shown figure decoding weight matrix dimension ten columns ten digits boolean matrix dimension represents corresponding value input digit example input digit represents training set values set since linear system weights found calculating decoding activity hidden neurons spike rate neuron measured desired range input values output neuron multiplied decoding weights since linear system weights found calculating moorepenrose penrose todd description one single digit training purposes used sample digits hence dimensions img vin change respectively use digits test dataset digits dimensions img vin change respectively testing phase predicted output product compared expected output obtain error rate number unrecognised digits among test digits address details testing section averaging output system linear sum weighted spike trains neurons algorithm theory methodology recognition classification handwritten digits standard machine learning problem form mnist database lecun become benchmark problem hence proof concept used proposed design framework implement digit recognition system figure importantly system could used pattern recognition applications mnist database digits represented pixels training testing dataset contain digits respectively system modelling aim develop fast hardware pattern recognition system running real time rather aiming lowest test error thus adopted hardwaredriven method implement system achieve best performance hardware resources method first consider hardware figure system topology inputs pixels connected hidden layer neurons using randomly weighted connections output layer consists linear neurons output layer weights solved analytically using pseudoinverse operation constraints building blocks optimised conversion reduce hardware cost significantly negligible performance loss presented detail section compare performance differences section conversion carried comparing grey scale value larger pixel set else set fpga implementations significant difference hardware cost implementations latter requires many digital signal processors dsps importantly number represented would lead huge data storage requirement would bottleneck system thus implemented system using numbers guarantee pixels digit input layer nonlinearly projected high dimensional hidden layer neuron hidden layer encoder first generate uniformly distributed random weight pixel one input digit sum weighted pixels generating stimulus verification hardware system random weights used software hardware models produce identical results software model random weights generated using special routines difficult implement hardware implementing design hardware modelled system python popular software programming language using representation ensure software hardware results avoid performance drop malfunctioning system hardware due conversion floating fixed point numbers models presented remaining part section software models unless otherwise specified one option use look table lut fpga store random weights generated software model major drawback solution requires significant amount memory scales linearly number input neurons hidden layer neurons fpga implementations efficient way generate random numbers use linear feedback shift registers lfsrs previously used implement randomly weighed connectivity spiking neural network wang based work developed encoder uses lfsrs perform nonlinear projection implemented input layer input layer read digits mnist database map input layer pixels one one task consists converting dimension also converting grey scale value number ranges pixels binary value latter major difference system existing algorithms tapson van schaik lecun else stim max end represents firing rate neuron result input stimulus represents index neuron neural core calculated shown different neurons represents size hidden layer represents maximum value stimulus stim represents current value input stimulus using integer range code input range figure shows tuning curves set proposed neurons using transfer function thus nonlinear function stimulus since value negative system requires stimulus nonlinearly encoded firing rate neuron hardware intensive use digital circuits implement conventional nonlinear functions tanh instead piecewise linear function easily implemented using single multiplier present implementation detail section figure tuning curves proposed neuron figure shows tuning curve neurons lfsr encoder software ensure random weights identical implementations highly optimised encoder hardware implementation details presented section rate neuron nef intrinsically uses spike rates calculate weights filters sum weighted output spikes implement desired function contrast implemented neurons neurons compute firing rate directly neurons implemented neurons fpga done previously wang average firing rates would measured value input stimulus compute decoding weights method quite inefficient inflexible would repeat measurements time parameters neurons change another drawback spiking neurons running real time would able accurately communicate firing rate short time period would significantly limit usage real time applications using neurons actual firing rate communicated immediately presenting stimulus neurons feature quite important applications need control interactive robotic systems hidden layer refer set neurons neural core used standard building block digit recognition system multiple neural cores easily combined build neural networks using design framework furthermore development cycle neural networks significantly shortened requirement measurement firing rate anymore since neural core set known tuning curves hidden layer implemented identical neural cores total neurons synaptic connections hidden layer size achieved best performance memory usage compare performance differences section given input image encoder generate via random weight projection different vin neuron core even core contains identical neurons words even though neuron neural core neuron neural core tuning curve function vin highly likely get different vin firing rates different system neurons system compute correctly neurons reproduce firing rate one used calculate decoding weights words computed firing rate must repeatable given input value based requirements proposed compute firing rate neuron using index array together stimulus value produce nonlinearity using following algorithm regression decoding weights obtained calculating pseudoinverse however matrix size requires huge amount memory computational time previously developed online pseudoinverse update method opium tapson van schaik incremental method compute pseudoinverse solution regression stim problem requires significantly less memory hence use method compute decoding weights chose use resolution decoding weights obtain best performance memory usage address details section pseudoinverse method gives best solution lowest square root error given matrix given set random weights necessarily achieve lowest test error mnist data set adopted regression method find best seed used encoder generate random weights turn change matrix way obtain lowest possible test error system figure shows flow regression method uses simplified version opium called opium lite tapson van schaik fast online method calculating approximation pseudoinverse significantly quicker opium find output weights resulting slightly worse test error opium lite used different random seeds different random weight vectors seed found target error desired threshold full scale opium used compute decoding weights seed guarantee opium lite able achieve target error desired threshold mechanism introduced system timeout activated regression run seeds happens simply use seed far resulted lowest error use full scale opium compute decoding weights hardware implementation topology efficiently implement system fpga use approach cassidy wang thakur leverages digital circuit fpgas easily run clock speed clock period thus exploit timemultiplexing approach simulate neurons powers two preferable optimise memory use storage millisecond implementing one physical neuron fpga refer neurons neurons means every clock cycle neuron processed neuron updated every resolution generally acceptable neural simulations approach however constrained data storage requirement sram limited size usually tens mbs due bandwidth constraints difficult use memory approach new values need available memory every clock cycle provide simulation furthermore architecture system complex using memory needs dedicated memory controller nevertheless using memory promises ability implement much larger networks investigate option future designs however chose use onchip memory current work keep architecture simple figure flow proposed regression method figure fpga implementation proposed system system topology internal structure timemultiplexed system figure shows topology fpga implementation system consists input layer encoder hidden layer neural cores output layer neurons encoder hidden layer implemented timemultiplexing approach figure shows internal structure consists physical encoder physical neuron global counter weight buffer global counter processes encoders neurons sequentially decoding weights physical neuron stored weight buffer simplicity let assume encoder neuron processed one clock cycle means every clock cycle encoder generate stimulus input digit corresponding neuron generate firing rate stimulus multiply decoding weights ten numbers ten digits obtained using opium input digit change remain static neurons finish processing output every neuron ten weighted firing rates accumulated corresponding output neuron using pipelined architecture result calculating one time step encoder neuron available turn encoder neuron comes around description assumes takes one clock cycle process one encoder neuron timing requirement quite difficult meet practical design address issue detail next section binary value saves significant hardware resources fpga since otherwise would need multipliers compute multiplication pixels corresponding random weights binary pixel used control multiplexer one connected corresponding random weight tied zero value pixel high corresponding random weight accumulated generation stimulus hidden layer neuron major challenge implementing encoder hardware using approach meet timing requirement need sum weighted pixels since neuron needs processed one clock cycle moreover operation require adders cost significant amount hardware resources introduction pipelines mitigate critical timing requirement need even adders compromise chose process encoder neuron time slot four clock cycles encoder perform sum operation four cycles sum weighted pixels modification mitigates critical timing requirement also reduces number adders needed price paid timemultiplexing rate divided four hence neurons rather neurons figure shows structure physical encoder consists input buffer global counter random weight generators implemented lfsr multiplexers sum module input digit arrives stored input buffer time slot global counter sends stored digits multiplexers generating weighted pixels lowest bits sent first clock cycle time slot higher bits next clock physical encoder encoder generate uniformly distributed random weight pixel input digit sum weighted pixels generate stimulus neuron hidden layer input digit converting value pixel figure structure physical encoder cycle one one highest bits fourth clock cycle implemented single multiplier computes multiplies one one decoding weight digit recognition system implemented neuron needs multiply ten decoding weights ten digits implementation would instantiate ten identical neurons one decoding weight output neuron would cost multipliers whole operation would require multiplications since time slot consists four clock cycles distribute multiplications four clock cycles multipliers needed based strategy neuron efficiently implemented three identical multipliers shown figure number implementable multipliers usually one bottlenecks generator generates random number divided four random signed numbers hence generators provide totally random weights sent corresponding multiplexer lfsrs reload initial seed obtained using pseudoinverse method arrival input digit keeps generating random numbers new input digit arrives way guarantee encoder generate exact set random weights incoming digit given seed fly generation scheme reduces usage memory significantly requirement storing random weights anymore seeds need stored accumulator module sums weighted pixels four clock cycles generating stimulus neuron naive implementation would need parallel adder create large delay mitigate critical timing requirement use pipeline consists fourteen parallel adders one parallel adder since pipelined design stimulus neuron still generated every time slot latency two clock cycles physical neuron rate neuron achieves significant reduction memory usage since computes firing rate index input stimulus fixed parameters none need memory access memory access needed read decoding weights previous work wang physical neuron already figure structure physical neuron table device utilisation altera cyclone adaptive logic modules alms rams dsps design multiplier inputs bits wide output result bits wide three multipliers need four clock cycles process algorithm multiplier first cycle computes represented number multiplying second cycle latches input multiplier third fourth cycle multiplies decoding weight respectively multiplier first second third fourth cycle multiplies decoding weight respectively multiplier first second third fourth cycle multiplies decoding weight respectively since pipelined design output neuron updated time slot latency four clock cycles output layer output layer consists ten neurons figure linearly sum results neurons since system sum accumulation outputs neurons time slot computational cost reduced magnitudes hence implementation output neuron need register adder neurons processed index output neuron maximum value sent result indicates likely input digit values ten output neurons cleared utilisation system developed using standard asic design flow thus easily implemented manufacturing technologies integrated circuit implementation desired design flow adopted designed verified module separately module level verification complete modules integrated together toplevel verification successfully implemented proposed neural cores yielding neurons altera cyclone fpga terasic cyclone starter kit design uses less hardware resources exception rams table note utilisation table includes circuits carry tasks jtag interface figure histogram error rate configuration configuration normalised histogram difference paired errors blue sample distributions modelling data red distribution estimated mean difference data system goal develop hardware system running real time rather exploiting algorithm accurate possible performance results obtained using full test set handwritten digits training full digit training set unless otherwise specified results presented section obtained using software python models results presented section obtained hardware implementation results results presented focus different design choices affect performance proposed comparison across different configurations compared previous work tapson van schaik made three major modifications greyscale pixel input images replaced black white binary pixels tanh neurons hidden layer replaced rate neurons numbers decoding weights replaced numbers investigated effects modifications using four configurations configuration configuration used previous work tapson van schaik configuration used black white images configuration used black white images rate neurons instead tanh neurons configuration three modifications hidden layer consisted neurons four configurations lite calculate decoding weights test error significantly reduces simulation time needed tests still providing fair comparison four configurations first investigated effect using binary values input layer compared performance result one using values binary values see figure top two panels show histogram number errors test patterns given skewed nature two error distributions rather simply reporting indicate statistical significance difference chosen display full distribution set random weight vectors used configuration determine paired difference two configurations shown histogram figure modelled distribution difference errors using distribution optimal modelling distributions approximately gaussian contain outliers followed bayesian estimation method according kruschke kruschke using markov chain monte carlo configuration test runs conducted different random seed set seeds used four configurations encoder generate random weights since goal exercise simply investigate impact three modifications performance rather find best possible performance used first five steps regression method used opium figure histogram error rate configuration normalised histogram difference paired errors blue sample distributions modelling data red distribution estimated mean difference data figure histogram error rate configuration normalised histogram difference paired errors blue sample distributions modelling data red distribution estimated mean difference data simulation simulated markov chain steps discarded first steps burn period figure shows distribution mean values distribution modelling data red curves figure show examples distribution parameters mean standard deviation normality parameter see kruschke taken random markov chain distribution mean value difference data figure see configuration results errors average define difference fewer errors region practical equivalence rope words consider insignificant change fewer errors tests change less note highest density interval hdi distribution mean difference errors outside rope therefore conclude changing input images grey scale binary values results small significant increase error around figure error rates function number neurons hidden layer respectively clear error decreases number hidden layer neurons although diminishing return since system used timemultiplexing approach rate neurons hardware cost single neuron almost negligible memory required decoding weights linearly proportional size hidden layer thus bottleneck system achieve good balance desired accuracy memory chose implement hidden layer rather neurons next investigated effect using rate neurons hidden layer distribution errors configuration configuration shown figure compared configuration figure paired difference shown figure figure shows distribution mean difference errors configuration configuration shows changing tanh neurons rate neurons increases number errors approximately however difference strongly significant hdi entirely outside rope indicating difference within region practical equivalence amongst possible mean values finally investigated effect using decoding weights figure shows distribution errors configuration difference configuration configuration close zero figure fact distribution mean error difference entirely within rope indicating somewhat surprisingly significant loss performance using output weights instead floating point weights system performance explore best performance proposed system achieve runs carried using full regression method figure different random seeds lowest error achieved lite full version opium respectively decoding weights obtained full version opium loaded fpga board real time digit recognition pixels input digits converted binary values software client software sent selected test digit fpga via jtag interface since system runs hidden layer contains neurons time slot four clock cycles processing time one input digit yielding digit recognitions per second due fact system used neurons one single neuron layer maximum number digit recognitions processed one neuron layer per second system used less hardware resources exception rams multiple neuron layers instantiated run parallel practical scale system process millions digit recognitions one second address details section performance drop configuration merely therefore conclude digit recognition system modifications made achieved significant reductions terms hardware cost minimal drop performance size hidden layer scenario used configuration previous section changed hidden layer size range neurons size ten test runs different random seed conducted reduce testing time used opium lite calculate decoding weights calculate test error discussion comparison solutions work reported constitutes basis building general purpose hardware pattern recognition systems using nef hence mainly interested scale median error runs figure hidden layer neurons performance hardware cost concentrate comparing work solutions developed similar goals rather solutions extremely optimised achieving lowest error rate mnist although efficiently implemented hardware table comparison solutions ibm truenorth system system building neural networks running real time merolla programmed digit recognition achieved result error rate test set mnist cores consisted spiking neurons needs bits memories esser hence system achieved much lower error rate significantly fewer hardware resources especially memories table regarding processing speed system needs time steps one process one digit whereas system needs approximately times speedup moreover system consists feature extractor clusters extracts features data system hence easily configured different input data without feature extractions truenorth system however much applications besides pattern recognition task compared system error computation time resources minitaur bits truenorth bits work bits mainly used encoders hence maximum number physical hidden neurons implemented memory requirement one single hidden neuron layer bits sram bits used implement hidden neuron layers scale system need use external memories bandwidth requirement indeed bottleneck approach new values need available memory every four clock cycles maximum theoretical bandwidth one sdram memory one sram memory board bits bits respectively memory general achieve efficiency theoretical bandwidth need flow control takes consideration bus turn around time refresh cycles maximum number neuron arrays adding ones using onchip sram theoretical maximum number neuron layers yielding neurons maximum number digit recognitions processed one neuron layer per second maximum number digit recognitions processed system parallel layers therefore per second minitaur neural network accelerator achieved error rate deep spiking network neurons neil liu since scheme used variant approach needs neurons physically implemented cost one single neuron also negligible bottleneck memory neuron used minitaur needs bits memories connection weight needs bit memories neuron needs bit memories decoding weights processing time minitaur one digit table approximately times slower system future work since larger scale pattern recognitions carried future work focus scaling network presented scalable design fully digital implementation number hidden neurons implemented single physical neuron increase linearly amount available memory long multiplexing scale keeps time resolution within biological time scale number physical neurons increase linearly number available alms programmability fpga especially decoding weights makes integration system desired pattern recognition applications seamless however advantages running networks strongly reduced neural networks take long time compute decoding weights hence another major improvement speed computationally extensive task one promising solution implement opium fpga since algorithm adaption procedure without requirement hundreds gigabyte rams quite friendly hardware implementation running opium real time makes possible upgrade system true turnkey solution pattern recognition real world addition since proposed system need feature extraction could used pattern recognition tasks speaker recognition natural language processing following calculation use digits recognition system metric different applications require different amounts hardware resources still using topology calculate theoretical maximum network size fpga board terasic board containing altera stratix fpga alms two sdrams four srams one single hidden layer requires alms acknowledgment tapson van schaik learning pseudoinverse solution network weights neural netw work supported australian research council grant support altera university program gratefully acknowledged work inspired capo caccia cognitive neuromorphic engineering workshop telluride neuromorphic workshop thakur hamilton tapson van schaik lyon fpga implementation car model cochlea ieee international symposium circuits systems vogelstein mallik vogelstein cauwenberghs dynamically reconfigurable silicon array spiking neurons synapses ieee trans neural netw references boahen neurogrid emulating million neurons cortex conf proc ieee eng med biol soc suppl wang cohen stiefel hamilton tapson van schaik fpga implementation polychronous spiking neural network delay adaptation front neurosci cassidy andreou georgiou design one million neuron single fpga neuromorphic system multimodal scene analysis annu conf inf sci wang hamilton tapson van schaik compact neural core digital implementation neural engineering framework eliasmith anderson neural engineering computation representation dynamics neurobiological systems boston mit press wang hamilton tapson van schaik compact reconfigurable implementation synaptic plasticity spiking neurons ieee international symposium circuits systems iscas ieee eliasmith stewart choo bekolay dewolf tang tang rasmussen model functioning brain science wang hamilton tapson van schaik fpga design framework spiking neural networks ieee international symposium circuits systems iscas melboune ieee esser andreopoulos appuswamy datta barch amir arthur cassidy flickner merolla cognitive computing systems algorithms applications networks neurosynaptic cores international joint conference neural networks ijcnn ieee wang hamilton tapson van schaik implementation polychronous spiking neural network delay adaptation front neurosci kruschke bayesian estimation supersedes test exp psychol lecun bottou bengio haffner learning applied document recognition proc ieee wang hamilton tapson van schaik neuromorphic implementation multiple synaptic plasticity rules neural networks front neurosci merolla arthur cassidy sawada akopyan jackson imam guo nakamura million integrated circuit scalable communication network interface science neil liu minitaur spiking network accelerator ieee trans large scale integr penrose todd generalized inverse matrices math proc cambridge philos soc pfeil jeltsch petrovici schmuker schemmel meier six networks universal neuromorphic computing substrate front neurosci tapson cohen afshar stiefel buskila wang hamilton van schaik synthesis neural networks spike pattern recognition processing front neurosci
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binomial edge ideals block graphs feb faryal chaudhry ahmet dokuyucu rida irfan bstract find class block graphs whose binomial edge ideals minimal regularity consequence characterize trees whose binomial edge ideals minimal regularity also show binomial edge ideal block graph depth initial ideal ntroduction paper study homological properties classes binomial edge ideals let simple graph vertex set let polynomial ring variables field set binomial edge ideal defined binomial edge ideals introduced algebraic homological properties binomial edge ideals studied several papers conjectured extremal betti numbers denotes lexicographic order induced conjecture proved cycles complete bipartite graphs shown closed graph regularity expressed combinatorial data graph recall graph closed quadratic basis respect lexicographic order support conjecture given show section block graph depth depth see theorem block graph mean chordal graph property two maximal cliques intersect one vertex also section show similar equality regularity precisely theorem show reg reg constitute subclass block graphs see section definition figure example shown connected graph vertex set reg length longest induced path mathematics subject classification key words phrases binomial edge ideals regularity depth first third author supported higher education commission pakistan abdus salam school mathematical sciences lahore pakistan main motivation work answer following question may characterize connected graphs whose longest induced path length reg succeeded answer question trees theorem show tree whose longest induced path length reg caterpillar caterpillar tree tree property contains path vertex either vertex adjacent vertex weakly closed graphs introduced class graphs includes closed graphs paper shown tree caterpillar weakly closed graph mind theorem theorem states reg connected closed graph whose longest induced path length computer experiments tempted formulate following conjecture connected weakly closed graph whose longest induced path length reg reliminaries section introduce notation used paper summarize results binomial edge ideals let simple graph vertex set loops multiple edges furthermore let field polynomial ring variables set binomial edge ideal associated generated quadratic binomials binomial edge ideals introduced papers first recall basic definitions graph theory vertex whose deletion graph gives graph connected components called cut point chordal graph graph without cycles length greater equal clique graph complete subgraph cliques graph form simplicial complex called clique complex facets maximal cliques graph block graph chordal every two maximal cliques one vertex common class considered theorem clique complex chordal graph property exists leaf order facets means facets may ordered every leaf simplicial complex generated leaf simplicial complex facet property exists another facet say every facet bases binomial edge ideals let lexicographic order induced natural order variables shown basis respect order may given terms admissible paths definition let two vertices path called admissible following conditions fulfilled either proper subset sequence path given admissible path set xik yil theorem set binomials admissible path rom reduced basis respect lexicographic order induced primary decomposition binomial edge ideals theorem shows particular radical monomial ideal implies binomial edge ideal radical well hence equal intersection minimal primes minimal prime ideals determined let possible empty subset let connected components denotes restriction vertex set complete graph vertex set let let theorem particular minimal primes among theorem let connected graph vertex set minimal prime ideal one set satisfies condition theorem called set nitial ideals binomial edge ideals block graphs section first show block graph connected components depth depth denotes lexicographic order induced ring begin following lemma lemma let graph vertex set let proof therefore show inclusion obvious since inclusion let take minimal generator obviously let means admissible path contain vertex follows path hence theorem let block graph depth depth number connected component proof let connected components depth depth depth moreover jgc thus depth depth depth jgc therefore without loss generality may assume connected theorem know depth order show depth proceed induction number maximal cliques let clique complex let leaf order facets simplex statement well known let since leaf exists unique vertex say branch let ftq facets intersect leaf vertex following proof theorem may write shown proof theorem follows obtained replacing cliques ftq clique vertex set also restriction vertex set lemma hence lemma get therefore get consequently following exact sequence using lemma thus actually following exact sequence since inherits properties smaller number maximal cliques follows inductive hypothesis depth depth let polynomial ring since graph vertices connected components satisfies conditions inductive hypothesis implies depth hence depth next observe obtained replacing clique vertex set clique vertex set hence inductive hypothesis depth since connected vertex set cardinality hence applying depth lemma exact sequence get depth depth let connected graph vertex set consists sequence maximal cliques dim together additional edges form intersection point two consecutive cliques vertex degree words obtained graph whose binomial edge ideal see theorem attaching edges intersection points facets therefore looks like graph displayed figure igure graph obviously property longest induced path length equal connected graph satisfies conditions say case dim called caterpillar graph also note chordal property two distinct maximal cliques intersect one vertex connected block graph theorem let vertex set reg reg proof let consists sequence maximal cliques condition add edges condition maximal cliques additional whiskers proceed induction number maximal cliques closed graph whose binomial edge ideal cohenmacaulay hence statement holds theorem let let leaf order facets obviously may choose leaf order arguments notation proof theorem get sequence observe hence inductive hypothesis reg reg graph two connected components one say possible component say occurs case clique dimension inductive hypothesis obtain reg reg reg reg relations yield reg exact sequence get max reg reg reg theorem know reg reg theorem reg using inequalities get desired conclusion inomial edge ideals caterpillar trees matsuda murai showed connected graph vertex set reg denotes length longest induced path conjectured reg line graph several recent papers concerned conjecture see example one may ask well characterize connected graphs whose longest induced path length reg section answer question trees caterpillar tree tree property contains path vertex either vertex adjacent vertex clearly caterpillar tree positive integer igure caterpillar igure induced graph caterpillar trees first studied harary schwenk graphs applications chemistry physics figure example caterpillar tree displayed note caterpillar tree narrow graph sense cox erskine conversely one may easily see narrow tree caterpillar tree moreover observed tree caterpillar graph weakly closed sense definition given next theorem characterize trees reg length longest induced path theorem let tree vertex set whose longest induced path length reg caterpillar proof let caterpillar tree whose longest induced path length definition caterpillar tree follows hence reg theorem conversely let reg assume caterpillar contains induced subgraph vertices figure theorem follows reg thus since reg reg see corollary follows reg contradiction hypothesis eferences conca gorenstein ladder determinantal rings london math soc cox erskine closed graphs dokuyucu extremal betti numbers classes binomial edge ideals applications caterpillar trees chemistry physics math chem ene herzog hibi binomial edge ideals nagoya math ene zarojanu regularity binomial edge ideals appear math nachr herzog hibi monomial ideals graduate texts mathematics springer herzog hibi hreinsdotir kahle rauh binomial edge ideals conditional independence statements adv appl math harary schwenk number caterpillars discrete math matsuda murai regularity bounds binomial edge ideals commut algebra matsuda weakly closed graphs binomial edge ideals arxiv ohtani graphs ideals generated commun algebra saeedi madani kiani binomial edge ideals graphs electron combin saeedi madani kiani regularity binomial edge ideals graphs arxiv zafar zahid betti numbers classes binomial edge ideals electron combin bdus alam chool athematical ciences niversity ahore uslim ahore pakistan address chaudhryfaryal faculty athematics omputer cience ovidius niversity amaia onstanta umina niversity outh ast urope olentina ucharest romania address bdus alam chool athematical ciences niversity ahore uslim ahore pakistan address ridairfan
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khor static dynamic characteristics protein contact networks susan abstract principles underlying protein folding remains one nature puzzles important practical consequences life approach gathered momentum since late looks protein heteropolymers folding process lens complex network analysis consequently body empirical studies describing topological characteristics protein contact networks linking topological characteristics protein folding present paper primarily review rich area delves deeper certain aspects emphasizing links suggests unconventional places may lurking within protein contact networks considers dynamical view protein contact networks closer scrutiny protein contact networks raises new questions research identifies new regularities may useful parameterize network approach protein folding preliminary experiments model confirm regularities identified easily reproduced random effects indeed grand challenge protein folding elucidate process generates specific diverse linkage patterns protein contact networks also reproduces dynamic behavior proteins fold keywords network analysis protein contact networks protein folding introduction breaking code underlying protein folding remained intellectually tantalizing puzzle well problem great practical significance everything protein requires correct folding normal circumstances appears embedded sequence afinsen although minority rely aid chaperone molecules fulfill destiny due large sizes sequences take random search approach protein folding deemed infeasible practical biological purposes levinthal however argument based separability protein folding problem problem separated parts solved independently assembled optimal conceived way levinthal paradox zwanzig karplus argument supported observation sections protein sequences propensity fold secondary structures medicine protein misfolding identified causative factor diseases cystic fibrosis als alzheimer chen proteins play many roles biological cells structural material contact info whimsical description see herbert simon parable two watchmakers architecture complexity sciences artificial mit press khor catalysts adaptors hormones transporters regulators tramontano proteins attain functionality unique though necessarily static native three dimensional states ultimate expression genes thus ability predict three dimensional structure protein sequence useful comprehend genomic data current approaches protein structure prediction problem comparative modeling methods sequence threading methods rely heavily existing knowledge protein sequences folds tramontano notwithstanding success methods understanding protein folding process first principles complete satisfying solution may prove invaluable protein design therapies targeting protein misfolding general protein folding process occurs stages essentially begins linear organized backbone protruding side chain groups obtains local structure form secondary alpha helices beta sheets finally global structure secondary structures arrange compactly three dimensions long time spontaneous biological attributed various physical forces chemical constraints impacting protein molecule however last decade another theory based network topology protein native state blossomed alm baker even suggest protein folding mechanisms landscapes largely determined topology native state relatively insensitive details interatomic theory network view protein molecules mostly native states adopted general recipe transform protein molecule network represent amino acid residues nodes contact spatial distances pairs amino acid residues certain threshold links protein contact networks maps vendruscolo constructed cartesian coordinates amino acid residues protein molecules stored protein data bank pdb berman examining protein contact networks pcn researchers compiled list topological characteristics shared diverse terms structural class homology taxon set proteins speculated reasons observed topological characteristics relation protein folding mechanism example common feature protein contact networks nature lattice like clustering coefficients random graph like diameters characteristic path lengths watts strogatz need rapid communication amino acid residues facilitate interaction cooperativity crucial protein folding frequently cited reason feature pcns vendruscolo dokholyan atilgan del sol protein contact networks also reported exhibit high assortativity values related protein folding speeds bagler sinha khor paper collected pcn structural characteristics reported literature various sets proteins using different methods pcn construction applied single set proteins dataset single pcn construction method find general reported network characteristics hold also carry line inquiry found literature green higman probing effects links links network characteristics pcns throughout paper investigations led new research questions new quantitative qualitative observations pcns could helpful finally realize network approach protein folding new observations include links like distribution link sequence distance proteins large maximum link sequence distance protein size however model network approach protein folding far complete problem choosing endpoints links remains elusive protein amino acids involved least one interaction protein contact map pcn protein contact network pcn vendruscolo adjacency matrix representing network whose nodes atom amino acid residues protein link placed pair nodes node pair situated less distance threshold apart distance node pairs euclidean distance cartesian coordinates obtained protein data bank pdb berman paper structural characteristics pcns seem overly sensitive choice threshold value bartoli build pcn protein dataset examples four pcns found figure proteins randomly chosen differences size nodes pcns respectively far referred pcns uniform entities actuality variations way pcns constructed example single pcn may represent several proteins rather single protein pcns may also represent different aspects surface core states native transitional structural classes types globular fibrous proteins nodes links pcns may carry different meanings atoms side chain group amino acid may included node may represent one atom multiple links nodes weighted links allowed examples alternative pcn constructions found references paper incidentally earlier use matrix depict protein folding found tanaka scheraga khor figure left protein contact networks four proteins dataset shaded cell denotes link nodes right networks produced two pcns top bottom links pink links turquoise yellow highlights nodes direct neighbours sections text provide explanation protein dataset proteins dataset table selected literature surveyed specifically green higman dataset encompass proteins different protein classes fold types branches life sizes range residues figure proteins form single component unusually high link density pcns constructed excluded dataset proteins nodes pcns dssp output kabsch sander reverse situation truncated dssp output another protein dataset described appendix used check increase confidence main observations made paper table proteins dataset number residues nodes khor proteins figure size pcns terms number atoms documented pdb files links link placed pair nodes representing atom amino acids node pair situated less apart section partitioned set links pcn two sets links links link nodes classified absolute distance amino acid sequence chain distance threshold links connect amino acids far apart primary structure close spatial proximity tertiary structure study green higman although also consider unless otherwise stated links refer links figures examine various aspects pcn links dataset similar observations made dataset appendix number links pcn increases linearly protein size number nodes pcn figure number links regardless also increases linearly protein size albeit slower rate figure comparison cut half sequence distance threshold links figures average links pcn mean standard deviation mean std dev instead entire pcn gaci balev considered subgraph induced set amino acids participating secondary structures links belonging less evolutionary conserved loop regions sses excluded call subgraph secondary structure elements interaction network gaci balev observed ratio number interactions secondary structures total number interactions exceed khor pcn links links count links nodes figure link count protein size proportion links links links nodes figure links proportion links protein size fraction links averages standard deviation average standard deviation pcn links lattice lattice link density nodes figure link density protein size khor pcns low link densities link density decreases rapidly protein size figure link density fraction actual links possible links plot link density pcn figure straightens line section reasons become clearer section figure compares link densities pcns two linear lattices latticev linear lattice nearest neighbours left right possible nodes two ends lattice chain fewer links rest nodes middle links given network size number nodes pcn link densities indistinguishable link densities figure node degree degree distribution typically node pcn represents atom amino acids section degree node number links attached node present context node degree measures number contacts node pcn gaci balev remarked homogeneity node degree mean degree increases slightly protein size falls within range absence nodes much higher degrees attributed excluded volume effect imposes physical limit number residues reside within given radius around another amino acid node degree distribution probability node particular degree node degree distributions variety pcns pcns constructed different ways different protein sets characteristic gaussian distribution green higman atilgan bagler sinha although green higman found node degree distributions pcns consider interactions exhibit tendencies exponential highly right skewed distributions irrespective fold type however pcn unlike pcn pcns referenced paper amino acid residue still represented single node green higman pcn however atoms amino acid considered deciding contact link establishment multiple links nodes allowed therefore green higman pcn register much larger node degrees reported modes green higman gaci balev also observed tendencies node degree distributions caveat sharp mean node degree finding connectivity patterns pcns attractive since connectivity pattern direct consequences efficiency information transfer robustness random attacks proteins green higman atilgan node degree statistics pcns presented mean node degrees independent protein size hover around mean node degree averaged pcns standard deviation little difference observed corresponding khor mean node degrees median node degrees standard deviation values four times smaller corresponding mean node degrees statistics point towards bell shaped degree distributions reported literature shown figure four proteins node degree max mid mean min nodes figure node degree summary statistics pcns average mean node degree standard deviation probability degree figure node degree distributions four pcns dataset since partition set links section similarly classify set nodes two subsets although two subsets necessarily disjoint set nodes endpoints links set nodes endpoints links figure compares sizes subsets find generally slightly smaller proteins entirely surprising since expect almost nodes found mean std dev mean std dev number nodes thus significant overlap almost subset khor relationship found mean std dev green higman found nodes within pcns contacts pcns mean std dev figure similar observations made dataset appendix node count nodes figure number nodes endpoints links number nodes endpoint links protein size fraction nodes averages standard deviation fraction nodes averages standard deviation proportion nodes nodes figure sizes different subsets nodes proportion nodes pcns average nodes nodes able distinguish protein without prior knowledge structure could important step predicting structure proteins unfortunately observed half amino acids protein sequence expected least one khor range interaction protein native state perhaps better strategy try identify distinguishing characteristics nodes involved interactions node degree characteristics use two ways comparison first compute node degree statistics nodes using node degree values pcn constructed result first comparison displayed figure yielded significant difference confidence level twosided paired test paired wilcoxon test nonetheless observed differences degree distributions pcn constructed pcn constructed narrow degree ranges pcns limits ability observe behavior several scales figure pcn avg pcn avg degree proteins figure degree statistics based respectively pcns constructed links links error bars mark standard deviation probability probability degree degree figure degree distributions pcns constructed links khor second compute node degree statistics nodes using node degree values entire pcn figure displays result second comparison according paired test paired wilcoxon test significant difference confidence level two vectors however given high overlap difference though statistically significant may slight magnitude degree avg avg proteins figure degree statistics based node degree values entire pcn error bars mark standard deviation clustering clustering transitivity reflects cliquishness nodes network node connects node node likely nodes connected different ways measuring transitive relationships strongly indicated graphs presence triangles one way taking average clustering nodes network yield clustering coefficient degree node number links exist amongst nodes watts strogatz vendruscolo reports values around pcn different construction bagler sinha found values around irrespective protein structural class defined scop murzin pcns values averaged std dev independent protein size figure compares values theoretical values regular cregular random crandom networks size number nodes crandom cregular average degree number nodes watts theoretical values regular networks cregular indistinguishable values surprising since mean degree used khor values pcns slightly smaller cregular closer values last observation telling includes protein backbone links protein backbone main source high clustering bartoli green higman observed pcns built links relatively lower values pcns links phenomenon observed proteins figure shows effect randomizing pcn links unlike randomizing links randle randomizing links randse reduces clustering levels similar random graphs randomizing links randall clustering coefficient regular pcn random nodes figure clustering coefficient values one standard deviation pcns clustering coefficients average standard deviation clustering coefficient pcn randle randse randall proteins figure randomizing reduces clustering randomizing links randomized usual manner maslov sneppen rewiring nodes preserving node degrees without introducing multiple links nodes example randomize set links randle two links picked uniformly khor random replacement already exist network replace high clustering pcns almost always discussed terms property watts strogatz since property also involves characteristic path length statistic section importance short distances protein folding inevitably dominates discussion result role high clustering local interaction level protein folding somewhat explained small exception bagler sinha suggest high clustering pcns indicative modular hierarchical organization result protein folding process called investigation hypothesis bagler sinha bagler sinha report negative correlation protein folding speed high clustering levels pcns take account links however find significant relationship protein folding speed clustering levels entire pcns hence exact role clustering plays protein folding still open question since clustering comes links comprise backbone secondary structure links may high clustering links necessary merely scaffolding device create contacts alternatively may high clustering levels found pcns side effect another network characteristic clearer relationship aspect protein folding example assortativity protein folding speed section could local organization links really play role aspect protein folding social networks transitive relationships assumed part nature human social behavior problem explain establishment subsequent efficient reuse random interesting example study kleinberg although far analogy human social networks protein contact networks kind residue interaction network applies protein folding remains seen assortativity network assortativity refers extent nodes associate connect kind common form assortativity measured pcns node degree section throughout paper positive assortativity refers proclivity nodes small large degree link nodes small large degree using method newman bagler sinha report correlation coefficients considered unusual networks biological origins discuss point ahead nonetheless positive assortativity values could correlated positive manner protein folding speeds bagler sinha similarly find pcns positive correlations figure assortativity values average standard deviation independent khor protein size figure uses alternative method less sensitive effects superconnected nodes assess degree assortativity method positive correlation degree average degree nodes directly neighbouring nodes degree interpreted evidence positive assortativity general plots figure agree summary figure although plots show weakening even reversal relationship node degree increases could due limits node degree bagler sinha coupled assortativity values close figure telling like clustering section assortativity sensitive random rewiring links randse random rewiring links randle figure analyzed separately links show less positive assortativity links means like clustering high positive assortativity values pcns also mainly stem assortativity pcn randall nodes figure pcns positive degree assortativity values close mean assortativity standard deviation average neighbourhood degree degree figure average neighbourhood degree degree average degree nodes adjacent nodes degree positive correlation implies positive assortativity khor assortativity pcn randle randse randall pcms figure effect randomizing links assortativity randse produces larger effect randle assortativity nodes figure assortativity links links previously number biological networks including gene regulatory networks described disassortative suggested negative assortativity specifically nodes high degree hubs directly connected networks studied advantageous sense effects harmful perturbations could better localized maslov sneppen brede sinha conclude stability networks assortative mixing node degree positive assortativity declines rapidly increases network size system fragile stability easily dislodged equilibrium state small disturbances however positive assortativity pcns may actually work benefit protein folding process helping information flow quickly thus facilitate coordinated action crucial correct rapid protein folding native state proteins adjust thus vulnerability perturbations marginal stability may advantageous taverna goldstein khor indeed percolation important vazquez moreno conclude node degree assortativity make networks robust random node edge removals additionally could delegation responsibilities interacting biological networks sense amino acid chain product gene translation concerns containing unwanted perturbations folding protein may taken care disassortative structure genetic networks additionally since less assortative figure may still degree isolation parts node subsets pcn buffer unexpected perturbations appears yet interesting discussions topic average path length average path length network average shortest path represents number links needs traversed average trying move nodes network pcns known average path lengths characteristic random graphs size respective protein sets pcn constructions vendruscolo report average path length bagler sinha report average path length path length widely mentioned topological feature pcns highly relevant protein folding vendruscolo dokholyan atilgan del sol argument short path lengths conducive rapid communication amino acid molecules protein facilitates interaction cooperativity crucial protein folding revisit point elsewhere manuscript submitted briefly find shorter distances guarantee finding global optimum easily shorter average path lengths result links fitness function additional links increase frustration figure gives various path length statistics pcns diameter maximum shortest path length found average path length increase logarithmically protein size nodes increases compared average path lengths canonical networks average path lengths pcns much shorter regular graphs approximate average path lengths expected random graphs size figure although standard deviations indicate rather large dispersions shortest path lengths pcns average path length median path length values close figure gather little skew distribution shortest path lengths pcns indeed case figure shows gaussian like distributions shortest path lengths four pcns khor path length maximum mean median nodes figure diameter average one standard deviation median path length pcns mean path length regular pcn random nodes figure average path lengths pcns much closer average path lengths random networks lrandom average path length regular networks lregular lrandom lregular average degree number nodes watts probability shortest path length figure distribution shortest path lengths four pcns khor figures respectively show effect randomizing links average path lengths diameters pcns like clustering section assortativity section randomizing links randle produces smaller effect randomizing links randse paired test paired wilcoxon test report significant difference confidence level randse randle vectors mean path length values figure indirectly shows larger proportion magnitude pcn average path length due mean path length pcn randle randse randall nodes pcn randle randse randall maximum path length figure effect randomizing links average path lengths pcns randse stronger effect randle proteins figure effect randomizing links diameters pcns randse stronger effect randle randomization links also reveals possible rearrange links pcn average path length significantly reduced thus question short distances important protein folding average path lengths pcns shorter khor preserving high clustering positive assortativity important protein folding adequate answer since still possible maintain clustering positive assortativity levels higher would random graphs significantly reducing average path lengths pcns randomizing links randle investigate question elsewhere find evidence showing randle less conducive search pcn manuscript submitted pcns recent history science complex networks distribution lack scale often associated node degree variable already touched upon section relation pcns section offer two degree places pcns distribution found first resides across proteins pcns relationship nodes link density derives directly mean node degree constant figure second resides within individual pcns terms distribution link sequence distances link density distribution across pcns figure left link density plot figure scale figure right similar plot figure appendix show clear relationships atypical exponent number links mean average node degree number nodes link density empirical data tells section link density link density pcn links pcn links nodes nodes figure links density plots logarithmic scales link sequence distance distribution pcn figure summarizes link sequence distance pcns sequence distance absolute distance link protein sequence section minimum sequence distance pcns find mean link sequence distance generally increases protein size consequence pulled maximum link sequence distance generally increases protein size however median link sequence distance remains fairly constant protein size khor well average link sequence distance per pcn similar observations made pcns dataset figure summary statistics hint distribution link sequence distances link sequence distance distributions four proteins scale seen figures straight line region negative slope sharp point plot revealing traces taking maximum link sequence distance fraction number nodes per pcn reveals average links maximum span figures link sequence distance avg std dev median nodes link sequence distance max nodes figure link sequence distance summary statistics pcns indicate right skewed distributions link sequence distance reversed cumulative probability khor link sequence distance figure link sequence distance distributions pcns logarithmic scales nodes figure maximum link sequence distance fraction number nodes pcn averages reversed cumulative probability link sequence distance figure link sequence distance distributions pcns logarithmic scales khor nodes figure maximum link sequence distance fraction number nodes pcn averages dynamic view section study effects adding links pcn already containing shortrange links assume links independent links shorter sequence distance form links longer sequence distance links added one time order link sequence distance pay attention order links sequence distance use random selection instead admittedly crude way simulating folding process protein captures essence monitor change time two network characteristics node betweeness average path length based literature surveyed expect number nodes exhibit distinctively higher node betweeness expect average path length decrease time average path length decrease given although occasional small increases possible interesting pertinent way general decrease happens betweeness node measure node centrality importance terms path traversal nodes network specifically betweeness node fraction shortest paths found node pairs pass node effort identify amino acids play key role protein folding process vendruscolo observed key residues often exhibit significantly higher node betweeness protein transition state however signal attenuates protein reaches native state due compactness hence looking node betweeness protein native state pcn likely produce false positives protein nucleation sites nonetheless topological network structure understanding protein folding process may help understand protein unique native conformation chosen vendruscolo dokholyan used topological characteristics protein conformation graphs pcn distinguish conformations belonging protein transition state ensemble tse cross khor energy barrier reach protein native state versus found posttransition conformations tse conformations nucleus formed likely reach native state like conformations tse conformations formed nucleus less likely reach native state sense average path length conformation graphs pcn significantly shorter average path length pcns words tightening structure proteins approach native conformations become compact figure shows protein folds using scheme described average node betweeness decreases slightly experiences phase transition resulting sudden substantial increase average node betweeness accompanied also increase node betweeness standard deviation distinct shortening average path length figure median maximum node betweeness also register sudden substantial increases point time figure effects indicate wider dispersion node betweeness values amongst nodes nodes exhibiting much higher node betweeness observed vendruscolo figure compare distribution node betweeness three points time transition point around transition point end folding process three plots highly use plot helps visualization limiting comparison three points time find definite widening node betweeness distribution followed narrowing node betweeness distribution tail plot swishes right left betweeness nodes become evenly spread amongst nodes native state however folding dynamics shows subtle transition figure right around transition point average path length shortens faster rate figure sudden substantial increase average node betweeness figure maximum node betweeness actually dipping rapidly figure spike median node betweeness figure effects indicative widening narrowing range node betweeness values figure convergence kind average median values get closer appears betweeness nodes become evenly spread amongst nodes native state therefore become less indicative key residues analogous idea crossing energy barrier protein folding kinetics change node betweeness could seen crossing kind communication barrier within nucleation model protein folding time prior transition point could regarded nucleation growth phase nucleus formed protein polymer becomes committed native conformation quickly reaches indicated rapid shortening average path lengths would interesting investigate effect test reliability median node khor betweeness indicator nucleation event protein folding monitor nucleation rates proteins different fold types sizes see whether constants specific avg median node betweeness avg median max median max time time shortest path length shortest path length median median maxim median time time maxim nodes betweeness relationships hpl hpl time time figure left plots right plots time step marks addition longrange link pcn pcn links average node betweeness standard deviation median node betweeness lie average values throughout median maximum node betweeness values average path length measured using harmonic mean method newman allow possibility disconnected pcns hpl reliable apl dealing disconnected networks figure khor reversed cumulative probability node betweeness reversed cumulative probability figure distribution node betweeness three points time logarithmic scales tail ends plots swishes right left node betweeness figure distribution node betweeness three points time logarithmic scales tail ends plots swishes one direction towards left comparison figures show average path length changes links added random order added order sequence distance proteins drop average path length occurs earlier links added random order links added order link sequence distance order kind decrease similar observed figure section hence protein folding specific rewiring process dokholyan link sequence distance link sequence distance khor time time hpl shortest path length shortest path length hpl time apl time hortest path length shortest path length time apl time figure behavior average path length links added nondecreasing order left versus random order right link sequence distance link sequence distance khor time shortest path length shortest path length time hpl hpl time time shortest path length shortest path length apl apl time time figure behavior average path length links added nondecreasing order left versus random order right network approach protein folding network perspective protein folding problem boils generating right set links amino acid residues protein many network characteristics pcns discovered past decade less published effort integrate knowledge single model use protein folding problem perhaps due current difficulty creating networks simultaneously fit number static dynamic structural characteristics still incomplete description pcns specific diverse arrangement links pcns even khor relative success protein structure prediction methods section outline approach protein folding uses several network structural characteristics pcns mentioned paper model still infancy hope good first step realizing network approach protein folding initio least put rest conceptions pcn models model begins links arranged linear lattice nodes use nodes node nearest neighbours left right possible gives approximately links since found section gives empirically reasonable number links start links added lattice endpoints link selected uniformly random set nodes absolute distance within section made two independent runs set statistics monitored link sequence distance distribution node degree distribution average path length clustering assortativity figure specifically plots present results experiment model results compared statistics pcn runs made model ring topology plots experiments ring topology initial lattice still linear links added lattice ring results topologies showed significant differences failed approximate statistics sorting links generated model plots help narrow gap although difference average path length behavior observed initial descend slower problem identifying pitfalls model lacks amongst elements reliable way identify endpoints longrange links set last observation section figure implies part links links connect nodes many contacts pcn point given assortative nature pcns section frequent description proteins domains modules protein folding strict process secondary tertiary structures beyond see messier view may tempting picture links connecting hub nodes nodes high degree however picture may entirely accurate two possibly related reasons first pcns proteins native state hubs necessarily occupy central position section correlation node degree node betweeness rapidly declines protein size increases figure appendix implies many shortest paths pcn khor hub nodes since links provide important links pcns contradictory reversed cumulative probability think range links connecting hub nodes link sequence distance degree time asso rtativity sterin efficien robability time time figure results network approach protein folding compared network statistics pcn size pcn clustering coefficient assortativity value nodes endpoints links model ring linear topology nodes ierarc trality rrelatio khor nodes nodes figure left pearson correlation node degree centrality pcns right gao hierarchy index pcns possible consequence also pcns actually show declining hierarchical structure increases protein size figure following trusina hierarchical organization gauged formulae gao defines hierarchical path one satisfies one following conditions monotonically increasing monotonically decreasing node degree monotonically increasing node degree monotonically decreasing node degree network high fraction shortest paths also hierarchical paths hierarchical index close signaling pronounced hierarchical organization however declining average clustering levels increases node degree also used detect hierarchical organization ravasz ravasz barabasi using method aftabuddin kundu report evidence hierarchical structure weighted pcn similar greene higman pcn also observe hierarchical tendencies four randomly chosen pcns figure evidently reconciliation performed two measures hierarchy second evidence depending scale used positive correlation node degree hydrophobicity alves martinez aftabuddin kundu implies hub nodes pcns tend hydrophobic hydrophobic residues shun water molecules generally lie core protein result residues come close proximity many residues thus gain hub status relationship hydrophobicity high node degree assortativity average path length needs study keeping mind relationship may differ transition versus native state proteins summary outlook main objective review paper integrate many network characteristics pcns reported literature understand respective roles also interplay attempt apply statistics network approach protein folding also khor made addition considered dynamic aspects pcns network statistics change protein folds model woefully simple hope provides good hints building better models integrating dynamic static aspects pcns one easily imagine several ways modify model adding constraints restrict set section use kind preferential selection mechanism fit link sequence distance distribution section node degree distribution section model could also extended include link deletion rewiring dokholyan observed changes node degrees pretransition states may also interesting extend approach section investigate dynamics pcns couples conformational landscape scala doye kovacs auer studies show important protein arrive final destination native state gets journey folding pathway also important chen light exposition paper network approach may vehicle address key aspects protein folding problem integrated way existence unique native structure folding pathway conformation space native structure protein folding rates information amino acid sequence propel attainment table summarize network structural statistics mentioned paper pcns data sets find general agreement qualitative quantitative terms observed network structural characteristics two data sets reported literature topic course existing network characteristics measurements ways constructing pcns need explored however could also useful devise measures geared towards capturing specific linkage patterns pcns table summary quantitative values mean standard deviation reported section statistic fraction links links sequence distance average node degree fraction nodes endpoints links fraction nodes endpoints links clustering assortativity maximum link sequence distance khor throughout paper maintained separation links better understand respective contributions structural characteristics pcns ultimately prize predict links pcns within level accuracy notwithstanding physical chemical geometrical constraints challenging combinatorial task even could identify accurately probability link chosen random set possible links endpoints includes links sequence distances less hence pure network model protein folding like one attempted section unlikely winner likely model could one incorporates multiple approaches given importance short distances residues section sparseness links pcns could appear paradoxical however model show general takes links reduce average path length significantly levels improved upon addition links figure according ngo marks links interactions actually increases computational complexity protein folding problem see also ngo thus links sufficient reduce average path length good enough level least effort albeit could taken evolution quite time figure optimal wirings strategy may well adopted proteins way paradox references aftabuddin kundu weighted unweighted network amino acids within protein physica aftabuddin kundu hydrophobic hydrophilic charged amino acids network within protein biophysical journal alm baker matching theory experiment protein folding curr opin struc biol alves martinez inferring topological features proteins amino acid residue networks physica anfinsen principles govern folding protein chains science atilgan akan baysal communication residues significance protein dynamics biophysics journal auer miller krivov dobson karplus vendruscolo importance metastable states free energy landscapes polypeptide chains physical review letters bagler sinha network properties protein structures physica statistics mechanics apps bagler sinha assortative mixing protein contact networks protein folding kinetics structural bioinformatics bartoli fariselli casadio effect backbone properties protein contact networks physical biology berman westbrook feng gilliland bhat weissig shindyalov bourne protein data bank nucleic acids research http brede sinha assortative mixing degree makes network unstable arxiv khor chen ding nie serohijos sharma wilcox yin dokholyan protein folding archives biochemistry biophysics del sol fujihashi amoros nussinov residues crucial maintaining short paths network communication mediate signaling proteins molecular systems biology dokholyan ding shakhnovich topological determinants protein folding pnas doye network topology potential energy landscape static network physical review letters gaci balev node degree distribution amino acid interaction networks proc computational structural bioinformatics workshop washington usa gao inferring autonomous system relationships internet transactions networking theoretical studies protein folding ann rev biophys bioeng greene higman uncovering network systems within protein structures journal molecular biology kabsch sander dictionary protein secondary structure pattern recognition hydrogenbonded geometrical features biopolymers http karplus levinthal paradox yesterday today folding design kleinberg navigation small world nature kovacs szalay csermely water molecular chaperones act weak links protein folding networks energy landscape punctuated equilibrium changes point towards game theory proteins febs letters levinthal fold graciously debrunner munck eds mossbauer spectroscopy biological systems proceedings meeting held allerton house monticello illinois university illinois press maslov sneppen specificity stability topology protein networks science murzin brenner hubbard chothia scop structural classification proteins database investigation sequences structures mol biol http newman assortative mixing networks physical review letters newman structure function complex networks siam review ngo marks computational complexity problem prediction protein engineering ngo marks karplus computational complexity protein structure prediction levinthal paradox merz grand eds protein folding problem tertiary structure prediction birkhauser boston vazquez vespignani dynamical correlation properties internet physical review letters ravasz somera mongru oltvai barabasi hierarchical organization modularity metabolic networks science ravasz barabasi hierarchical organization complex networks physical review rost twilight zone protein sequence alignments protein engineering http sep scala amaral networks conformation space lattice polymer chain europhysics letters tanaka scheraga model protein folding incorporation ising model model proc natl acad sci usa taverna goldstein proteins marginally stable proteins tramontano protein structure prediction concepts applications verlag gmbh trusina maslov minnhagen sneppen hierarchy measures complex networks physical review letters vazquez moreno resilience damage graphs degree correlation physical review khor vendruscolo dokholyan paci karplus view amino acids play key role protein folding physical review watts strogatz collective dynamics networks nature watts small worlds dynamics networks order randomness princeton university press princeton zwanzig szabo bagchi levinthal paradox proc natl acad sci usa appendix protein dataset protein dataset extracted list unique chains archived eva rost proteins selected random list overlap pcns proteins constructed selected manner described sections producing valid pcns listed table work table pid proteins dataset number residues nodes proteins figure size pcns terms number atoms documented pdb files khor pcn links count nodes link density figure link count protein size proteins pcn links nodes figure link density protein size proteins link count nodes figure number links protein size fraction links fraction links khor node count nodes figure number nodes endpoints links nodes endpoint links protein size fraction nodes fraction nodes proportion nodes nodes figure sizes different subsets nodes proportion nodes pcns average nodes nodes khor node degree max mid mean min nodes figure node degree summary statistics pcns mean node degree average std dev probability degree figure node degree distributions four pcns dataset respective pcns nodes respectively clustering coefficient proteins figure clustering coefficients standard deviation mean std dev khor assortativity proteins figure assortativity values mean std dev link sequence distance avg std dev median nodes link sequence distance max nodes figure link sequence distance summary statistics pcns indicate right skewed distributions link sequence distance khor appendix additional material figure section figure centrality scatter plots four pcns correlation coefficients given table methods show weakening positive correlation increase report spearman rho values significance could computed due ties correlation method pearson kendall tau correlation coefficients significant confidence level average node centrality degree figure node centrality nodes degree averaged reported standard deviations relationship marginally positive grows weaker increase protein size large standard deviations point wide dispersion node centrality values seen scatter plots figure khor average clustering coefficient degree figure evidence hierarchical organization pcns
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existence automorphisms order finite thin apr marco ruscitti leire legarreta abstract paper study existence least one noninner automorphism order finite thin whenever prime marco ruscitti disim degli studi dell aquila aquila italy address leire legarreta matematika saila euskal herriko unibertsitatea bilbao spain address introduction main goal paper contribute longstanding conjecture berkovich posed conjectures every finite admits automorphism order denotes prime number problem conjecture attracted attention many mathematicians last couple decades confirmed many classes finite remarkable put record liebeck proved existence automorphism order finite class odd prime however fact always exists automorphism order finite class proved abdollahi conjecture confirmed finite regular schmid indeed deaconescu proved finite strongly frattinian moreover abdollahi proved finite powerful jamali visesh finite cyclic commutator subgroup realm finite groups quite recently result confirmed benmoussa guerboussa nilpotency mathematics subject classification key words phrases finite automorphisms derivation thin first author would like thank department mathematics university basque country excellent hospitality part paper written second author supported spanish government grants basque government grant class abdollahi ghoraishi wilkens finally abdollahi proved conjecture coclass contribution paper add another class finite list proving mentioned conjecture holds true finite thin whenever prime organization paper follows section exhibit preliminary facts tools used proof main result paper section recall elementary matters thin prove main result well throughout paper notation standard found instance preliminaries section recall facts derivations multiplicative setting related lemmas useful prove main theorem paper reader could referred details explicit proofs derivations definition let group let right derivation function terms properties derivation uniquely determined values set generators let free group generated finite subset let group whose free presentation normal closure set relations standard argument shows acts trivially indeed denote canonical homomorphism action given continuing notation following results lemma let every function extends unique way derivation lemma let let derivation given composition derivation conversely derivation defines uniquely derivation following lemma study relationship derivations automorphisms finite lemma let finite let normal abelian subgroup viewed derivation define uniquely endomorphism furthermore automorphism order reduce calculations terms commutators keep mind following result automorphisms order finite thin lemma let free group prime number derivation proof let since mod first assertion follows let prove second assertion induction clearly assertion holds let suppose inductive hypothesis let take let suppose berkovich conjecture finite thin develop section start enumerating structural properties concerned thin firstly let recall group antichain set mutually incomparable elements lattice normal subgroups maximal class lattice normal subgroups consists maximal subgroups terms lower central series thus maximal class one antichain consists maximal subgroups necessity extend family groups maximal class bigger family bound antichains leads introduce formal definition thin let introduce definition thin forwards mention well already proved results existence automorphisms order certain specific cases avoid repetitions definition let finite thin every antichain contains elements following results finite thin discussed lemma let finite thin let odd prime normal subgroup term lower central series unique normal subgroup order maximal class exponent lemma let finite thin let take following called coverty property holds remark properties shown lemma give lot information thin since elementary abelian thin order see every finite thin two generator group secondly know lower upper central series kind groups coincide consequently quotients series elementary abelian order consequently must cyclic order particular quotients lower upper central finite thin exponent moreover finite thin property stated lemma holds equivalently terms upper central series since liebeck proved existence least automorphism order finite class odd prime abdollahi proved existence automorphism order finite class abdollahi ghoraishi wilkens case finite nilpotency class study deal finite nilpotency class hand since deaconescu proved existence least automorphism order finite strongly frattinian may assume finite thin interested strongly frattinian words groups interest satisfy result due abdollahi nite automorphisms order leaving elementwise fixed thus view previous result order prove existence automorphism order finite thin may assume without loss generality condition holds thus using consequences got remark isomorphic elementary abelian group order also showed see theorem every finite thin group maximal class thus following focus attention finite thin odd prime ready prove next theorem theorem let finite thin odd prime automorphism order proof let denote nilpotency class said assume remark consequences assumptions know two generator lower upper central series coincide particular elementary abelian subgroup indeed order class lemma may assume exponent extraspecial exponent order goal obtain least automorphism order firstly define assignment generators free group generated two elements sending lemma possible extend assignments derivation secondly show map preserves relations defining quotient apply lemma induce derivation finally lift automorphisms order finite thin found map derivation applying lemma following paragraphs describe detail mentioned steps let assignment generators two generator free group lemma assignment extends uniquely derivation making perhaps abuse notation assume corresponds presentation let see equalities hold firstly let show trivial powers elements fact considering canonical epimorphism secondly let analyze behaviour commutators since holds applying previous equality get consequently taking also account get element moreover take account previous two equalities obtained well applying properly item lemma lemma induce derivation map defined law derivation lemma induces automorphism law leaving particular elementwise fixed clearly fact previous inclusion holds since allows prove automorphism order fact thus previous construction produce set automorphisms order whose size equal words whose size number possible choices images generators distinguish two possible cases former case produce automorphisms order however number inner automorphisms induced elements thus counting argument enough say automorphism order get statement theorem case otherwise latter case choose assignment central let automorphism order obtained assignment hand know inner exists element circumstances since would deduce contradiction analogous coverty property lemma consequently second case happen statement theorem proved acknowledgements first author would like thank department mathematics university basque country excellent hospitality part paper written also wish thank professors gustavo alcober norberto gavioli carlo maria scoppola suggestions references abdollahi finite class automorphisms order algebra abdollahi powerful automorphism order cohomology algebra abdollahi ghoraishi wilkens finite class automorphisms order beitr algebra geom abdollahi ghoraishi guerboussa reguiat wilkens automorphism order finite coclass group theory benmoussa guerboussa properties bull austral math soc brandl caranti scoppola metabelian thin math caranti mattarei newman scoppola thin groups prime powerorder thin lie algebras math deaconescu automorphisms order finite journal algebra brandl dilworth number subgroup lattices arch math gavioli pannone frobenius partitions extraspecial geometriae dedicata jamali existence automorphism order two finite bull aust math soc liebeck outer automorphisms nilpotent class london math soc robinson course theory groups second edition new york schmid cohomological property regular math unsolved problems group theory kourovka notebook edited mazurov khukhro russian academy sciences siberian division institute mathematics novosibirisk disim degli studi dell aquila aquila italy address matematika saila euskal herriko unibertsitatea bilbao spain address
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reassembling trees traveling salesman jens vygen jan university bonn revised version december abstract many recent approximation algorithms variants traveling salesman problem asymmetric tsp graph tsp tsp exploit fact solution natural linear programming relaxation written convex combination spanning trees main argument randomly sampling tree distribution completing tree tour minimum cost yields better approximation guarantee simply taking minimum cost spanning tree algorithm argue additional step help reassembling spanning trees sampling exchanging two edges pair spanning trees improve properties certain conditions demonstrate usefulness metric tsp devising deterministic algorithm improves previously best approximation ratio keywords traveling salesman problem tsp approximation algorithm spanning tree introduction traveling salesman problem tsp probably combinatorial optimization problem although general metric tsp algorithm approximation ratio still unbeaten seen progress several variants special cases particular since see vygen detailed survey many recent approximation algorithms begin solving natural linear programming relaxation proposed dantzig fulkerson johnson observed held karp solution scaling factor except case written convex combination equivalently probability distribution spanning trees course distribution far unique asadpour oveis gharan saberi singh improved approximation ratio asymmetric tsp graph tsp respectively randomly sampling spanning tree maximum entropy distribution describing completing tour optimal way kleinberg shmoys considered metric tsp showed spanning tree randomly chosen distribution describing good enough improve algorithm problem paper propose modify distribution sampling exchanging two edges pair trees certain properties obtain two new trees hence new distribution call step reassembling trees certain conditions two new trees better properties old ones tsp show step indeed improve approximation ratio tsp let metric tsp formally classical version tsp want visit set cities minimum total cost however rather returning origin end given origin destination input precisely given set two elements symmetric distance function satisfying triangle inequality throughout paper denote numberpof elements ask sequence minimized classical metric tsp special case note variants either require every city visited exactly equivalently drop requirement triangle inequality allow visiting cities tsp clearly easier classical metric tsp reduce latter former guessing two cities adjacent optimum tour particular unless approximation algorithm better ratio karpinski lampis schmied assume henceforth compute connected vertex set exacty odd degree graph called given eulerian walk using every edge exactly linear time shortcut whenever vertex visited time yields vertex set triangle expensive previous approximation algorithms algorithm originally designed classical metric tsp works also tsp computes minimum cost spanning tree complete graph spanned adds minimum cost set vertices whose degree wrong parity even odd vertices result hoogeveen showed algorithm approximation ratio tsp fact ratio asymptotically attained set examples kleinberg shmoys proposed christofides algorithm tsp proved approximation ratio algorithm quite simple computes optimum solution natural relaxation see writes convex combination spanning trees spanning trees computes minimum weight set vertices whose degree wrong parity obtains outputs best details subsection improvement algorithm applied general metrics improved analysis obtaining approximation ratio describe subsection special case metric closure unweighted graph better approximation algorithms obtained svensson mucha kleinberg shmoys order best known approximation ratio obtained vygen matches integrality ratio special case gao gave simpler proof result papers apply also generalization general see section brief discussion notation given instance let let edge set complete graph denotes set edges exactly one endpoint set edges endpoints write even set odd even cut odd intersection always contains odd number edges edmonds proved minimum weight computed polynomial time vector write moreover denotes incidence vector denote set edge sets spanning trees set odd even contains vertices whose degree wrong parity algorithm extension christofides algorithm proposed kleinberg shmoys therefore describe algorithm detail algorithm begins solving relaxation min subject even odd obviously integral solutions precisely incidence vectors edge sets hamiltonian indeed relaxation following idea karp kleinberg shmoys observed polytope hence contained spanning tree polytope edmonds optimum solution fact every feasible solution therefore written distribution theorem assume less spanning trees optimum solution spanning trees numbers computed polynomial time shown ellipsoid method schrijver using technique genova williamson henceforth also distribution rest introduction modify later algorithm computes minimum weight considers output best note trying leads deterministic algorithm least good randomly picking probability basic analysis follow basic analysis kleinberg shmoys cost algorithm computes given instance depends choice bomc min min note result edmonds johnson says minimum weight minimum polyhedron therefore bomc set vectors polyhedron analysis lies appropriate set vectors let call correction vectors although kleinberg shmoys used term meaning bound cost parity correction narrow cuts let elements called narrow cuts narrow cuts except valid correction vector leading bomc wolsey analysis algorithm classical metric tsp tsp general narrow cuts following property useful lemma kleinberg shmoys narrow cuts form chain sets proof hence suppose contradiction remark proposition given set narrow cuts computed polynomial time proof lemma implies order narrow cut form thus compute minimum done applications algorithm computing tree similarly lemma lemma proof let moreover kleinberg shmoys observed even even one every narrow cut note follow directly also get even many figure spanning tree filled circles show vertices edge set partitioned turquoise red narrow cuts shown grey dotted solid even number edges shown solid grey contains least one red least one blue edge let denote edge set let note see figure example observed fact every intersects every narrow cut fact every contradicting connected correction vectors consider nonnegative vector satisfying even lemma kleinberg shmoys every every proof let hence intersection odd intersection odd even hence due polyhedron distribution nonnegative vectors get bound bomc question choose vectors analyses kleinberg shmoys kleinberg shmoys also gao chose max even vectors recall edge set choice implies even required writing get bomc max even even max max used last inequality three papers choose vectors kleinberg shmoys showed chosen observing setting obtained bomc chose indeed due using observing one gets bomc setting yields bomc set yields bound otherwise used correction vectors gao simply chose incidence vector cheapest edge clearly best possible framework could obtain better approximation ratio new approach use ideas kleinberg shmoys see section like boundp constant fraction precisely vectors immediately yield bomc question large choose appendix give example shows general possible choose hence directly improve approximation ratio therefore modify reassembling trees contribute convex combination eliminate critical show section complete calculation section obtain improved approximation ratio new correction vectors since want bound weighted sum costs vectors multiple weighted sum costs pair make contribution vectors tree analysis two types contributions pair contribute total vectors trees latter contribution distributed follows narrow cut one cut contribute even analysis essentially equivalent choosing obtain improvement choosing individual values new vectors choose edge esc moreover let minimum cost edge pair choose number later set max even term direct contribution edges second term exactly still needed obtain even see proof lemma show total cost weighted sum second terms see section lemma vectors defined bomc proof note vectors nonnegative every even required bound immmediately follows using try maximize bounding cost cost vectors course depend choice desired inequality implied max even may seen formally shown proof lemma write inequality compact form see divide use following notation definition given define numbers even min otherwise need show enough every narrow cut lemma let even bomc proof every even min even compute max even even min even esc used esc second third inequality assertion follows lemma let quickly check obvious although need simply choose even otherwise even peven one peven inequality follows second one follows even fact obtain papproximation ratio tight one peven total let give informal description idea approximation guarantee readily improved cuts one peven see previous paragraph would like obtain critical cuts achieve reduce less critical cuts still enough critical cut esc belong critical cut choose even describe exactly beginning section case get however work critical cut every edge esc also belongs another critical cut case choose get note least one edge however due lemma many edges intersection two critical cuts certainly less one per tree average constant consider cuts called kleinberg shmoys prefer call cuts critical let adjacent cuts directions precisely max min lemma let critical cut lemma must trees less two edges disjoint union therefore every tree either larger exactly least two edges get leads improvement essentially argument seen figure however addition trees two edges four orange purple property reassembling step described next aims avoiding two four one two orange ones one two purple ones turn enough reassembling trees psection show reassemble trees contribute convex combination order remove certain bad given constant exact value chosen later number cuts left right note moreover type figure cut adjacent cuts types according shown right see one two edges two edge esc belongs shown top critical get least esc belongs green get strictly possible critical cut blue two edges possible average orange purple problematic even edges three edges called good shown types trees cuts type spanning tree cut remark type depend value problem since constant chosen later definition fix constant tree cut define type follows let type good otherwise type lmr note type good see figure list types good figure types tree figure assuming narrow cuts types also listed table page good figure spanning tree figure assuming narrow cuts grey vertical lines list type according cut marked good reassembling lemma key lemma reassembling trees lemma let let type type two edges moreover new trees type old trees type type type type type type type good see figure example proof let edge type index belongs cuts neither let belongs type good must edge edge cut edge must belong exists index moreover graph contains circuit circuit even intersection cut circuit contain however must contain least second edge besides contains figure reassembling trees exchanging two edges trees types lemma belongs circuit property obvious nothing changes note property follows show prove claim type good end observe let right endpoint right endpoint figure let observe crosses every cut among odd number times every cut even number times therefore contains neither contains even number edges contains even number edges let moreover contains odd number edges suppose otherwise type good guarantees get odd implies type good claim proved claim directly implies let suppose types since identical remaining reason types thus new type good suppose types identical remaining reason types thus contains odd number edges must contain implies contains another edge addition hence therefore new type resulting types let type every type apply previous lemma left right cut long possible order obtain algorithm round integer multiples small positive constant corollary constants algorithm given instance optimum solution set numbers computes another set numbers min proof compute cuts proposition process left right ignoring initially check types trees whenever two trees type respectively set min decrease increase chosen lemma type type good maintain properties min type good type good remove tree drops zero note stage integer multiples never trees symmetry also corollary constants algorithm given instance optimum solution set numbers computes another set numbers min conclude theorem constants algorithm given instance optimum solution computes set trees exists distribution min good proof first compute distribution less trees trees using ellipsoid method schrijver set technique genova williamson set set trees note let get min apply corollary applying corollary yields min distribution min required output note algorithm needs set contains less trees proof shows also compute distribution polynomial time needed analysis improved approximation ratio show set numbers total according large constant slightly larger determined later let lemma average trees narrow cut ideally would like even may impossible greater therefore cut precisely two numbers otherwise min max odd otherwise min max even set otherwise set less critical cuts lemma proof choice even let even one even pone even even get even even need analyze remaining cuts precisely hence critical cuts consider cuts subsection establish outlined lemma let let even moreover esc esc proof first let esc moreover esc let even min observe hence consider second term moreover esc holds particular esc bound critical cuts course using distribution according theorem lemma let let let distribution let many proof let adjacent cuts left right let note using lemma analogously odd thus even thus distinguish four cases case pgood let sphas type type good otherwise note trees lemma table hence using case taking weighted sum implies using many implies obtain case pgood let sphas type type good otherwise note trees lemma table type good good good number edges case case case case table types contributions bounds second column follow lemma use reassembling except fraction trees two orange types occur simultaneously two purple types occur simultaneously compensated good types hence using case taking weighted sum implies many implying case pgood let sphas type type good otherwise note trees lemma table hence using case taking weighted sum implies many case pgood symmetric case proved cases setting constants obtain main result easily theorem distribution obtained theorem bomc particular algorithm tsp integrality ratio proof use lemma need show let narrow cut shown lemma let narrow cut note thus particular apply lemma constants chosen holds values hence therefore using moreover even many peven together directly implies enhancements constants previous section optimal close bounds almost tight case many however suggest two ideas analysis leads improvement firstly since contrast analysis worst case occur slightly larger value one increase hence improve approximation ratio increasing slightly secondly analysis case lemma analysis tight case anyway indicate consider critical cut case assume simplicity many let let next cut left fraction trees larger others generally fraction trees see figure performed necessary calculations obtain best possible approximation ratio ideas however seems resulting improvements rather small much better bound would probably need stronger reassembling results discussion theorem readily leads also improved approximation ratio tsp simply applying theorem kleinberg shmoys see applications type figure analysis case theorem improves best known upper bound integrality ratio best known lower bound shown metric closure unweighted circuit metric closures unweighted graphs integrality ratio indeed even vygen proved general metrics remains open natural open question course improve approximation ratio improvement small reassembling technique could powerful able prove seems stronger version lemma would needed would also interesting generalize algorithm problem general approximation algorithm previous algorithm cheriyan friggstad gao work also general problem finally applying technique tsp variants would interesting acknowledgement author thanks three referees careful reading excellent suggestions references kleinberg shmoys improving algorithm path tsp journal acm article asadpour goemans oveis gharan saberi log log log approximation algorithm asymmetric traveling salesman problem proceedings annual symposium discrete algorithms soda cheriyan friggstad gao approximating connected algorithmica analysis new heuristic traveling salesman problem technical report graduate school industrial administration university pittsburgh dantzig fulkerson johnson solution large scale traveling salesman problem operations research edmonds chinese postman problem bulletin operations research society america edmonds submodular functions matroids certain polyhedra combinatorial structures applications proceedings calgary international conference combinatorial structures applications guy hanani sauer eds gordon breach new york edmonds johnson matching euler tours chinese postman mathematical programming gao algorithm path graph traveling salesman problem operations research letters gao metric path traveling salesman problem siam journal discrete mathematics genova williamson experimental evaluation algorithm traveling salesman problem algorithms esa lncs bansal finocchi eds springer berlin schrijver ellipsoid method consequences combinatorial optimization combinatorica hassin khuller raghavachari approximation algorithms bounded performance guarantees clustered traveling salesman problem algorithmica held karp problem minimum spanning trees operations research hoogeveen analysis heuristic paths cycles operations research letters karpinski lampis schmied new inapproximability bounds tsp algorithms computation proceedings isaac symposium lncs cai cheng lam eds springer svensson approximating graphic tsp matchings proceedings annual symposium foundations computer science focs mucha graphic tsp theory computing systems oveis gharan saberi singh randomized rounding approach traveling salesman problem proceedings annual ieee symposium foundations computer science focs eight approximation tsp paths integer programming combinatorial optimization proceedings ipco conference lncs correa goemans eds springer vygen shorter tours nicer ears path version subgraphs combinatorica vygen new approximation algorithms tsp optima wolsey heuristic analysis linear programming branch bound mathematical programming study appendix example following example shows get matter choose numbers represent solution arbitrary way convex combination spanning trees show one instance vertices edges obtain sequence instances extending part middle inserting copies blue part adding vertices edges step number next edge feasible solution grey vertical lines show narrow cuts consider red sets following singletons call set sets vectors easily shown linearly independent since constraints correspond sets equality indeed vertex polytope hence optimum solution objective function wills show bad incidence vectors trees following four trees spanning trees part grey cuts left top four trees types four trees edge within part belongs one narrow cut one narrow cut matter choose total edge cuts average narrow cut part receives shows reassembling trees necessary unable prove algorithm better approximation ratio obtain better ratio completely analysis would necessary fact better lower bound known
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